Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 219:43021d2b8756
separate cardinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 07 Aug 2019 09:50:51 +0900 |
| parents | OD.agda@eee983e4b402 |
| children | afc864169325 |
| rev | line source |
|---|---|
| 16 | 1 open import Level |
| 219 | 2 module cardinal where |
| 3 | 3 |
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separete constructible set
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4 open import zf |
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fce60b99dc55
posturate OD is isomorphic to Ordinal
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5 open import ordinal |
| 219 | 6 open import logic |
| 7 open import OD | |
| 23 | 8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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9 open import Relation.Binary.PropositionalEquality |
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separete constructible set
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10 open import Data.Nat.Properties |
| 6 | 11 open import Data.Empty |
| 12 open import Relation.Nullary | |
| 13 open import Relation.Binary | |
| 14 open import Relation.Binary.Core | |
| 15 | |
| 219 | 16 open OD.OD |
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posturate OD is isomorphic to Ordinal
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parents:
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17 |
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fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
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18 open Ordinal |
| 120 | 19 open _∧_ |
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22d435172d1a
separate logic and nat
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20 open _∨_ |
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separate logic and nat
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21 open Bool |
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fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
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parents:
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22 |
| 219 | 23 ------------ |
| 24 -- | |
| 25 -- Onto map | |
| 26 -- def X x -> xmap | |
| 27 -- X ---------------------------> Y | |
| 28 -- ymap <- def Y y | |
| 29 -- | |
| 30 record Onto {n : Level } (X Y : OD {n}) : Set (suc n) where | |
| 31 field | |
| 32 xmap : (x : Ordinal {n}) → def X x → Ordinal {n} | |
| 33 ymap : (y : Ordinal {n}) → def Y y → Ordinal {n} | |
| 34 ymap-on-X : {y : Ordinal {n} } → (lty : def Y y ) → def X (ymap y lty) | |
| 35 onto-iso : {y : Ordinal {n} } → (lty : def Y y ) → xmap ( ymap y lty ) (ymap-on-X lty ) ≡ y | |
| 51 | 36 |
| 219 | 37 record Cardinal {n : Level } (X : OD {n}) : Set (suc n) where |
| 38 field | |
| 39 cardinal : Ordinal {n} | |
| 40 conto : Onto (Ord cardinal) X | |
| 41 cmax : ( y : Ordinal {n} ) → cardinal o< y → ¬ Onto (Ord y) X | |
| 151 | 42 |
| 219 | 43 cardinal : {n : Level } (X : OD {suc n}) → Cardinal X |
| 44 cardinal {n} X = record { | |
| 45 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) | |
| 46 ; conto = onto | |
| 47 ; cmax = cmax | |
| 48 } where | |
| 49 cardinal-p : (x : Ordinal {suc n}) → ( Ordinal {suc n} ∧ Dec (Onto (Ord x) X) ) | |
| 50 cardinal-p x with p∨¬p ( Onto (Ord x) X ) | |
| 51 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 52 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
| 53 onto-set : OD {suc n} | |
| 54 onto-set = record { def = λ x → {!!} } -- Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X } | |
| 55 onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X | |
| 56 onto = record { | |
| 57 xmap = xmap | |
| 58 ; ymap = ymap | |
| 59 ; ymap-on-X = ymap-on-X | |
| 60 ; onto-iso = onto-iso | |
| 61 } where | |
| 62 -- | |
| 63 -- Ord cardinal itself has no onto map, but if we have x o< cardinal, there is one | |
| 64 -- od→ord X o< cardinal, so if we have def Y y or def X y, there is an Onto (Ord y) X | |
| 65 Y = (Ord (sup-o (λ x → proj1 (cardinal-p x)))) | |
| 66 lemma1 : (y : Ordinal {suc n}) → def Y y → Onto (Ord y) X | |
| 67 lemma1 y y<Y with sup-o< {suc n} {λ x → proj1 ( cardinal-p x)} {y} | |
| 68 ... | t = {!!} | |
| 69 lemma2 : def Y (od→ord X) | |
| 70 lemma2 = {!!} | |
| 71 xmap : (x : Ordinal {suc n}) → def Y x → Ordinal {suc n} | |
| 72 xmap = {!!} | |
| 73 ymap : (y : Ordinal {suc n}) → def X y → Ordinal {suc n} | |
| 74 ymap = {!!} | |
| 75 ymap-on-X : {y : Ordinal {suc n} } → (lty : def X y ) → def Y (ymap y lty) | |
| 76 ymap-on-X = {!!} | |
| 77 onto-iso : {y : Ordinal {suc n} } → (lty : def X y ) → xmap (ymap y lty) (ymap-on-X lty ) ≡ y | |
| 78 onto-iso = {!!} | |
| 79 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X | |
| 80 cmax y lt ontoy = o<> lt (o<-subst {suc n} {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} | |
| 81 (sup-o< {suc n} {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
| 82 lemma : proj1 (cardinal-p y) ≡ y | |
| 83 lemma with p∨¬p ( Onto (Ord y) X ) | |
| 84 lemma | case1 x = refl | |
| 85 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 86 |
| 218 | 87 func : {n : Level} → (f : Ordinal {suc n} → Ordinal {suc n}) → OD {suc n} |
| 88 func {n} f = record { def = λ y → (x : Ordinal {suc n}) → y ≡ f x } | |
| 217 | 89 |
| 218 | 90 Func : {n : Level} → OD {suc n} |
| 91 Func {n} = record { def = λ x → (f : Ordinal {suc n} → Ordinal {suc n}) → x ≡ od→ord (func f) } | |
| 92 | |
| 93 odmap : {n : Level} → { x : OD {suc n} } → Func ∋ x → Ordinal {suc n} → OD {suc n} | |
| 94 odmap {n} {f} lt x = record { def = λ y → def f y } | |
| 95 | |
| 219 | 96 lemma1 : {n : Level} → { x : OD {suc n} } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal {suc n} → Ordinal {suc n}) → ¬ ( x ≡ od→ord (func f) )) |
| 97 lemma1 = {!!} | |
| 218 | 98 |
| 219 | 99 |
| 100 ----- | |
| 101 -- All cardinal is ℵ0, since we are working on Countable Ordinal, | |
| 102 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 103 |
| 104 | |
| 105 |