Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison OD.agda @ 219:43021d2b8756
separate cardinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 07 Aug 2019 09:50:51 +0900 |
parents | eee983e4b402 |
children | 2e1f19c949dc |
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218:eee983e4b402 | 219:43021d2b8756 |
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620 ¬¬X∋x nn = not record { | 620 ¬¬X∋x nn = not record { |
621 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) | 621 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) |
622 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) | 622 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) |
623 } | 623 } |
624 | 624 |
625 ------------ | |
626 -- | |
627 -- Onto map | |
628 -- def X x -> xmap | |
629 -- X ---------------------------> Y | |
630 -- ymap <- def Y y | |
631 -- | |
632 record Onto {n : Level } (X Y : OD {n}) : Set (suc n) where | |
633 field | |
634 xmap : (x : Ordinal {n}) → def X x → Ordinal {n} | |
635 ymap : (y : Ordinal {n}) → def Y y → Ordinal {n} | |
636 ymap-on-X : {y : Ordinal {n} } → (lty : def Y y ) → def X (ymap y lty) | |
637 onto-iso : {y : Ordinal {n} } → (lty : def Y y ) → xmap ( ymap y lty ) (ymap-on-X lty ) ≡ y | |
638 | |
639 record Cardinal {n : Level } (X : OD {n}) : Set (suc n) where | |
640 field | |
641 cardinal : Ordinal {n} | |
642 conto : Onto (Ord cardinal) X | |
643 cmax : ( y : Ordinal {n} ) → cardinal o< y → ¬ Onto (Ord y) X | |
644 | |
645 cardinal : {n : Level } (X : OD {suc n}) → Cardinal X | |
646 cardinal {n} X = record { | |
647 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) | |
648 ; conto = onto | |
649 ; cmax = cmax | |
650 } where | |
651 cardinal-p : (x : Ordinal {suc n}) → ( Ordinal {suc n} ∧ Dec (Onto (Ord x) X) ) | |
652 cardinal-p x with p∨¬p ( Onto (Ord x) X ) | |
653 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
654 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
655 onto-set : OD {suc n} | |
656 onto-set = record { def = λ x → {!!} } -- Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X } | |
657 onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X | |
658 onto = record { | |
659 xmap = xmap | |
660 ; ymap = ymap | |
661 ; ymap-on-X = ymap-on-X | |
662 ; onto-iso = onto-iso | |
663 } where | |
664 -- | |
665 -- Ord cardinal itself has no onto map, but if we have x o< cardinal, there is one | |
666 -- od→ord X o< cardinal, so if we have def Y y or def X y, there is an Onto (Ord y) X | |
667 Y = (Ord (sup-o (λ x → proj1 (cardinal-p x)))) | |
668 lemma1 : (y : Ordinal {suc n}) → def Y y → Onto (Ord y) X | |
669 lemma1 y y<Y with sup-o< {suc n} {λ x → proj1 ( cardinal-p x)} {y} | |
670 ... | t = {!!} | |
671 lemma2 : def Y (od→ord X) | |
672 lemma2 = {!!} | |
673 xmap : (x : Ordinal {suc n}) → def Y x → Ordinal {suc n} | |
674 xmap = {!!} | |
675 ymap : (y : Ordinal {suc n}) → def X y → Ordinal {suc n} | |
676 ymap = {!!} | |
677 ymap-on-X : {y : Ordinal {suc n} } → (lty : def X y ) → def Y (ymap y lty) | |
678 ymap-on-X = {!!} | |
679 onto-iso : {y : Ordinal {suc n} } → (lty : def X y ) → xmap (ymap y lty) (ymap-on-X lty ) ≡ y | |
680 onto-iso = {!!} | |
681 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X | |
682 cmax y lt ontoy = o<> lt (o<-subst {suc n} {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} | |
683 (sup-o< {suc n} {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
684 lemma : proj1 (cardinal-p y) ≡ y | |
685 lemma with p∨¬p ( Onto (Ord y) X ) | |
686 lemma | case1 x = refl | |
687 lemma | case2 not = ⊥-elim ( not ontoy ) | |
688 | |
689 func : {n : Level} → (f : Ordinal {suc n} → Ordinal {suc n}) → OD {suc n} | |
690 func {n} f = record { def = λ y → (x : Ordinal {suc n}) → y ≡ f x } | |
691 | |
692 Func : {n : Level} → OD {suc n} | |
693 Func {n} = record { def = λ x → (f : Ordinal {suc n} → Ordinal {suc n}) → x ≡ od→ord (func f) } | |
694 | |
695 odmap : {n : Level} → { x : OD {suc n} } → Func ∋ x → Ordinal {suc n} → OD {suc n} | |
696 odmap {n} {f} lt x = record { def = λ y → def f y } | |
697 | |
698 | |
699 ----- | |
700 -- All cardinal is ℵ0, since we are working on Countable Ordinal, | |
701 -- Power ω is larger than ℵ0, so it has no cardinal. | |
702 | |
703 | |
704 |