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

comparison OD.agda @ 219:43021d2b8756

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separate cardinal
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 07 Aug 2019 09:50:51 +0900
parents eee983e4b402
children 2e1f19c949dc
comparison
equal deleted inserted replaced
218:eee983e4b402 219:43021d2b8756
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