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comparison OD.agda @ 365:7f919d6b045b

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 18 Jul 2020 12:29:38 +0900
parents 67580311cc8e
children f74681db63c7
comparison
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364:67580311cc8e 365:7f919d6b045b
325 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) 325 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
326 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) 326 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
327 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) 327 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
328 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy 328 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy
329 329
330 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
331 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
332 Replace : HOD → (HOD → HOD) → HOD
333 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x }
334 ; odmax = rmax ; <odmax = rmax<} where
335 rmax : Ordinal
336 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))
337 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
338 rmax< lt = proj1 lt
339 Union : HOD → HOD
340 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) }
341 ; odmax = osuc (od→ord U) ; <odmax = umax< } where
342 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U)
343 umax< {y} not = lemma (FExists _ lemma1 not ) where
344 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x
345 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y))
346 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U
347 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U))
348 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y)
349 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
350 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U)
351 lemma not with trio< y (od→ord U)
352 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
353 lemma not | tri≈ ¬a refl ¬c = <-osuc
354 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
355 _∈_ : ( A B : HOD ) → Set n
356 A ∈ B = B ∋ A
357
358 OPwr : (A : HOD ) → HOD
359 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) )
360
361 Power : HOD → HOD
362 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
363 -- ｛_｝ : ZFSet → ZFSet
364 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly
365
366 data infinite-d : ( x : Ordinal ) → Set n where
367 iφ : infinite-d o∅
368 isuc : {x : Ordinal } → infinite-d x →
369 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
370
371 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
372 -- We simply assumes infinite-d y has a maximum.
373 --
374 -- This means that many of OD may not be HODs because of the od→ord mapping divergence.
375 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x).
376 --
377 postulate
378 ωmax : Ordinal
379 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
380
381 infinite : HOD
382 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
383
384 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD
385 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
386 u : (y : Ordinal ) → HOD
387 u y = Union (ord→od y , (ord→od y , ord→od y))
388 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
389 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y))
390 lemma8 = ho<
391 --- (x,y) < next (omax x y) < next (osuc y) = next y
392 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y)
393 lemmaa {x} {y} x<y = subst (λ k → od→ord (x , y) o< k ) (sym nexto≡) (subst (λ k → od→ord (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
394 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y))
395 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
396 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y))
397 lemma9 = lemmaa (c<→o< (case1 refl))
398 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y))
399 lemma71 = next< lemma81 lemma9
400 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y))))
401 lemma1 = ho<
402 --- main recursion
403 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
404 lemma {o∅} iφ = x<nx
405 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1))
406
407 ω<next-o∅ : ({x : HOD} → od→ord x o< next (odmax x)) → {y : Ordinal} → infinite-d y → y o< next o∅
408 ω<next-o∅ ho< {y} lt = <odmax (infinite' ho<) lt
409
410 nat→ω : Nat → HOD
411 nat→ω Zero = od∅
412 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y))
413
414 ω→nat : (n : HOD) → infinite ∋ n → Nat
415 ω→nat n = lemma where
416 lemma : {y : Ordinal} → infinite-d y → Nat
417 lemma iφ = Zero
418 lemma (isuc lt) = Suc (lemma lt)
419
420 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n))
421 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) {!!} iφ
422 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) {!!} (isuc ( ω∋nat→ω {n}))
423
424 _=h=_ : (x y : HOD) → Set n
425 x =h= y = od x == od y
426
427 infixr 200 _∈_
428 -- infixr 230 _∩_ _∪_
429
430 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
431 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
432 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
433
434 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
435 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
436 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
437
438 empty : (x : HOD ) → ¬ (od∅ ∋ x)
439 empty x = ¬x<0
440
441 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
442 o<→c< lt = record { incl = λ z → ordtrans z lt }
443
444 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
445 ⊆→o< {x} {y} lt with trio< x y
446 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
447 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
448 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
449 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
450
451 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
452 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
453 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
454 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
455 union← X z UX∋z = FExists _ lemma UX∋z where
456 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
457 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
458
459 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
460 ψiso {ψ} t refl = t
461 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
462 selection {ψ} {X} {y} = record {
463 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
464 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
465 }
466 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y))))
467 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt )
468 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
469 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where
470 lemma : def (in-codomain X ψ) (od→ord (ψ x))
471 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
472 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
473 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
474 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
475 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
476 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
477 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
478 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
479 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
480 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
481
482 ---
483 --- Power Set
484 ---
485 --- First consider ordinals in HOD
486 ---
487 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
488 --
489 --
490 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
491 ∩-≡ {a} {b} inc = record {
492 eq→ = λ {x} x<a → record { proj2 = x<a ;
493 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
494 eq← = λ {x} x<a∩b → proj2 x<a∩b }
495 --
496 -- Transitive Set case
497 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
498 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
499 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) )
500 --
501 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
502 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
503 lemma refl (lemma1 lemma-eq )where
504 lemma-eq : ((Ord a) ∩ t) =h= t
505 eq→ lemma-eq {z} w = proj2 w
506 eq← lemma-eq {z} w = record { proj2 = w ;
507 proj1 = odef-subst {_} {_} {(Ord a)} {z}
508 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
509 lemma1 : {a : Ordinal } { t : HOD }
510 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t
511 lemma1 {a} {t} eq = subst (λ k → od→ord ((Ord a) ∩ k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
512 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a)))
513 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t)))
514 lemma : od→ord ((Ord a) ∩ (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord ((Ord a) ∩ (ord→od x)))
515 lemma = sup-o< _ lemma2
516
517 --
518 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
519 -- then replace of all elements of the Power set by A ∩ y
520 --
521 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
522
523 -- we have oly double negation form because of the replacement axiom
524 --
525 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
526 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
527 a = od→ord A
528 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
529 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t
530 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
531 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
532 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
533 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
534 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
535 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
536
537 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
538 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
539 a = od→ord A
540 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
541 lemma0 {x} t∋x = c<→o< (t→A t∋x)
542 lemma3 : OPwr (Ord a) ∋ t
543 lemma3 = ord-power← a t lemma0
544 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
545 lemma4 = let open ≡-Reasoning in begin
546 A ∩ ord→od (od→ord t)
547 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
548 A ∩ t
549 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
550 t
551 ∎
552 sup1 : Ordinal
553 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x)))
554 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A)))
555 lemma9 = <-osuc
556 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1
557 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9
558 lemmad : Ord (osuc (od→ord A)) ∋ t
559 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt)))
560 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A)
561 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
562 lemmaf : {x : Ordinal} → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x
563 lemmaf {x} lt = proj1 lt
564 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x
565 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt }
566 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A))
567 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac)
568 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t)
569 lemma7 with osuc-≡< lemmad
570 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
571 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where
572 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x
573 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t))
574 diso
575 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt )))
576 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
577 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t
578 lemmai = let open ≡-Reasoning in begin
579 od→ord (Ord (od→ord A))
580 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩
581 od→ord (Ord (od→ord t))
582 ≡⟨ sym diso ⟩
583 od→ord (ord→od (od→ord (Ord (od→ord t))))
584 ≡⟨ sym eq1 ⟩
585 od→ord (ord→od (od→ord t))
586 ≡⟨ diso ⟩
587 od→ord t
588 ∎
589 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
590 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A))
591 lemmak = let open ≡-Reasoning in begin
592 od→ord (ord→od (od→ord (Ord (od→ord t))))
593 ≡⟨ diso ⟩
594 od→ord (Ord (od→ord t))
595 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩
596 od→ord (Ord (od→ord A))
597 ∎
598 lemmaj : od→ord t o< od→ord (Ord (od→ord A))
599 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt
600 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))
601 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))))
602 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 )
603 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
604 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
605 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
606 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A )))
607
608
609 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
610 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
611 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
612 lemma lt y<x with osuc-≡< lt
613 lemma lt y<x | case1 refl = c<→o< y<x
614 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
615
616 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
617 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
618
619 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
620 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
621 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
622
623 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
624 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
625 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
626
627 infinity∅ : infinite ∋ od∅
628 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
629 lemma : o∅ ≡ od→ord od∅
630 lemma = let open ≡-Reasoning in begin
631 o∅
632 ≡⟨ sym diso ⟩
633 od→ord ( ord→od o∅ )
634 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
635 od→ord od∅
636 ∎
637 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
638 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
639 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
640 ≡ od→ord (Union (x , (x , x)))
641 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
642
643 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
644 isZF = record {
645 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
646 ; pair→ = pair→
647 ; pair← = pair←
648 ; union→ = union→
649 ; union← = union←
650 ; empty = empty
651 ; power→ = power→
652 ; power← = power←
653 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
654 ; ε-induction = ε-induction
655 ; infinity∅ = infinity∅
656 ; infinity = infinity
657 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
658 ; replacement← = replacement←
659 ; replacement→ = λ {ψ} → replacement→ {ψ}
660 -- ; choice-func = choice-func
661 -- ; choice = choice
662 }
663
330 HOD→ZF : ZF 664 HOD→ZF : ZF
331 HOD→ZF = record { 665 HOD→ZF = record {
332 ZFSet = HOD 666 ZFSet = HOD
333 ; _∋_ = _∋_ 667 ; _∋_ = _∋_
334 ; _≈_ = hod→zf._=h=_ 668 ; _≈_ = _=h=_
335 ; ∅ = od∅ 669 ; ∅ = od∅
336 ; _,_ = _,_ 670 ; _,_ = _,_
337 ; Union = hod→zf.Union 671 ; Union = Union
338 ; Power = hod→zf.Power 672 ; Power = Power
339 ; Select = hod→zf.Select 673 ; Select = Select
340 ; Replace = hod→zf.Replace 674 ; Replace = Replace
341 ; infinite = hod→zf.infinite 675 ; infinite = infinite
342 ; isZF = hod→zf.isZF 676 ; isZF = isZF
343 } where 677 }
344 module hod→zf where 678
345 ZFSet = HOD -- is less than Ords because of maxod 679
346 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
347 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
348 Replace : HOD → (HOD → HOD) → HOD
349 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x }
350 ; odmax = rmax ; <odmax = rmax<} where
351 rmax : Ordinal
352 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))
353 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
354 rmax< lt = proj1 lt
355 Union : HOD → HOD
356 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) }
357 ; odmax = osuc (od→ord U) ; <odmax = umax< } where
358 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U)
359 umax< {y} not = lemma (FExists _ lemma1 not ) where
360 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x
361 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y))
362 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U
363 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U))
364 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y)
365 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
366 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U)
367 lemma not with trio< y (od→ord U)
368 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
369 lemma not | tri≈ ¬a refl ¬c = <-osuc
370 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
371 _∈_ : ( A B : ZFSet ) → Set n
372 A ∈ B = B ∋ A
373
374 OPwr : (A : HOD ) → HOD
375 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) )
376
377 Power : HOD → HOD
378 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
379 -- ｛_｝ : ZFSet → ZFSet
380 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly
381
382 data infinite-d : ( x : Ordinal ) → Set n where
383 iφ : infinite-d o∅
384 isuc : {x : Ordinal } → infinite-d x →
385 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
386
387 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
388 -- We simply assumes infinite-d y has a maximum.
389 --
390 -- This means that many of OD may not be HODs because of the od→ord mapping divergence.
391 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x).
392 --
393 postulate
394 ωmax : Ordinal
395 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
396
397 infinite : HOD
398 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
399
400 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD
401 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
402 u : (y : Ordinal ) → HOD
403 u y = Union (ord→od y , (ord→od y , ord→od y))
404 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
405 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y))
406 lemma8 = ho<
407 --- (x,y) < next (omax x y) < next (osuc y) = next y
408 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y)
409 lemmaa {x} {y} x<y = subst (λ k → od→ord (x , y) o< k ) (sym nexto≡) (subst (λ k → od→ord (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
410 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y))
411 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
412 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y))
413 lemma9 = lemmaa (c<→o< (case1 refl))
414 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y))
415 lemma71 = next< lemma81 lemma9
416 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y))))
417 lemma1 = ho<
418 --- main recursion
419 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
420 lemma {o∅} iφ = x<nx
421 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1))
422
423 nat→ω : Nat → HOD
424 nat→ω Zero = od∅
425 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y))
426
427 ω→nat : (n : HOD) → infinite ∋ n → Nat
428 ω→nat n = lemma where
429 lemma : {y : Ordinal} → infinite-d y → Nat
430 lemma iφ = Zero
431 lemma (isuc lt) = Suc (lemma lt)
432
433 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n))
434 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) {!!} iφ
435 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) {!!} (isuc ( ω∋nat→ω {n}))
436
437 _=h=_ : (x y : HOD) → Set n
438 x =h= y = od x == od y
439
440 infixr 200 _∈_
441 -- infixr 230 _∩_ _∪_
442 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
443 isZF = record {
444 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
445 ; pair→ = pair→
446 ; pair← = pair←
447 ; union→ = union→
448 ; union← = union←
449 ; empty = empty
450 ; power→ = power→
451 ; power← = power←
452 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
453 ; ε-induction = ε-induction
454 ; infinity∅ = infinity∅
455 ; infinity = infinity
456 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
457 ; replacement← = replacement←
458 ; replacement→ = λ {ψ} → replacement→ {ψ}
459 -- ; choice-func = choice-func
460 -- ; choice = choice
461 } where
462
463 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
464 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
465 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
466
467 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
468 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
469 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
470
471 empty : (x : HOD ) → ¬ (od∅ ∋ x)
472 empty x = ¬x<0
473
474 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
475 o<→c< lt = record { incl = λ z → ordtrans z lt }
476
477 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
478 ⊆→o< {x} {y} lt with trio< x y
479 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
480 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
481 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
482 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
483
484 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
485 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
486 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
487 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
488 union← X z UX∋z = FExists _ lemma UX∋z where
489 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
490 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
491
492 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
493 ψiso {ψ} t refl = t
494 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
495 selection {ψ} {X} {y} = record {
496 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
497 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
498 }
499 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y))))
500 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt )
501 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
502 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where
503 lemma : def (in-codomain X ψ) (od→ord (ψ x))
504 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
505 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
506 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
507 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
508 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
509 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
510 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
511 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
512 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
513 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
514
515 ---
516 --- Power Set
517 ---
518 --- First consider ordinals in HOD
519 ---
520 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
521 --
522 --
523 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
524 ∩-≡ {a} {b} inc = record {
525 eq→ = λ {x} x<a → record { proj2 = x<a ;
526 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
527 eq← = λ {x} x<a∩b → proj2 x<a∩b }
528 --
529 -- Transitive Set case
530 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
531 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
532 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) )
533 --
534 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
535 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
536 lemma refl (lemma1 lemma-eq )where
537 lemma-eq : ((Ord a) ∩ t) =h= t
538 eq→ lemma-eq {z} w = proj2 w
539 eq← lemma-eq {z} w = record { proj2 = w ;
540 proj1 = odef-subst {_} {_} {(Ord a)} {z}
541 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
542 lemma1 : {a : Ordinal } { t : HOD }
543 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t
544 lemma1 {a} {t} eq = subst (λ k → od→ord ((Ord a) ∩ k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
545 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a)))
546 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t)))
547 lemma : od→ord ((Ord a) ∩ (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord ((Ord a) ∩ (ord→od x)))
548 lemma = sup-o< _ lemma2
549
550 --
551 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
552 -- then replace of all elements of the Power set by A ∩ y
553 --
554 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
555
556 -- we have oly double negation form because of the replacement axiom
557 --
558 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
559 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
560 a = od→ord A
561 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
562 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t
563 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
564 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
565 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
566 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
567 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
568 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
569
570 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
571 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
572 a = od→ord A
573 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
574 lemma0 {x} t∋x = c<→o< (t→A t∋x)
575 lemma3 : OPwr (Ord a) ∋ t
576 lemma3 = ord-power← a t lemma0
577 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
578 lemma4 = let open ≡-Reasoning in begin
579 A ∩ ord→od (od→ord t)
580 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
581 A ∩ t
582 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
583 t
584 ∎
585 sup1 : Ordinal
586 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x)))
587 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A)))
588 lemma9 = <-osuc
589 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1
590 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9
591 lemmad : Ord (osuc (od→ord A)) ∋ t
592 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt)))
593 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A)
594 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
595 lemmaf : {x : Ordinal} → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x
596 lemmaf {x} lt = proj1 lt
597 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x
598 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt }
599 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A))
600 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac)
601 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t)
602 lemma7 with osuc-≡< lemmad
603 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
604 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where
605 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x
606 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t))
607 diso
608 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt )))
609 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
610 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t
611 lemmai = let open ≡-Reasoning in begin
612 od→ord (Ord (od→ord A))
613 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩
614 od→ord (Ord (od→ord t))
615 ≡⟨ sym diso ⟩
616 od→ord (ord→od (od→ord (Ord (od→ord t))))
617 ≡⟨ sym eq1 ⟩
618 od→ord (ord→od (od→ord t))
619 ≡⟨ diso ⟩
620 od→ord t
621 ∎
622 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
623 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A))
624 lemmak = let open ≡-Reasoning in begin
625 od→ord (ord→od (od→ord (Ord (od→ord t))))
626 ≡⟨ diso ⟩
627 od→ord (Ord (od→ord t))
628 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩
629 od→ord (Ord (od→ord A))
630 ∎
631 lemmaj : od→ord t o< od→ord (Ord (od→ord A))
632 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt
633 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))
634 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))))
635 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 )
636 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
637 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
638 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
639 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A )))
640
641
642 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
643 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
644 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
645 lemma lt y<x with osuc-≡< lt
646 lemma lt y<x | case1 refl = c<→o< y<x
647 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
648
649 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
650 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
651
652 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
653 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
654 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
655
656 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
657 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
658 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
659
660 infinity∅ : infinite ∋ od∅
661 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
662 lemma : o∅ ≡ od→ord od∅
663 lemma = let open ≡-Reasoning in begin
664 o∅
665 ≡⟨ sym diso ⟩
666 od→ord ( ord→od o∅ )
667 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
668 od→ord od∅
669 ∎
670 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
671 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
672 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
673 ≡ od→ord (Union (x , (x , x)))
674 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
675
676
677 Union = ZF.Union HOD→ZF
678 Power = ZF.Power HOD→ZF
679 Select = ZF.Select HOD→ZF
680 Replace = ZF.Replace HOD→ZF
681 infinite = ZF.infinite HOD→ZF
682 isZF = ZF.isZF HOD→ZF
683