Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 224:afc864169325
recover ε-induction
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 10 Aug 2019 12:31:25 +0900 |
| parents | 43021d2b8756 |
| children | 5f48299929ac |
| rev | line source |
|---|---|
| 16 | 1 open import Level |
| 224 | 2 open import Ordinals |
| 3 module cardinal {n : Level } (O : Ordinals {n}) where | |
| 3 | 4 |
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separete constructible set
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5 open import zf |
| 219 | 6 open import logic |
| 224 | 7 import OD |
| 23 | 8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 224 | 9 open import Relation.Binary.PropositionalEquality |
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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 | |
| 224 | 16 open inOrdinal O |
| 17 open OD O | |
| 219 | 18 open OD.OD |
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posturate OD is isomorphic to Ordinal
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19 |
| 120 | 20 open _∧_ |
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separate logic and nat
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21 open _∨_ |
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22 open Bool |
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od→lv : {n : Level} → OD {n} → Nat
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23 |
| 219 | 24 ------------ |
| 25 -- | |
| 26 -- Onto map | |
| 27 -- def X x -> xmap | |
| 28 -- X ---------------------------> Y | |
| 29 -- ymap <- def Y y | |
| 30 -- | |
| 224 | 31 record Onto (X Y : OD ) : Set n where |
| 219 | 32 field |
| 224 | 33 xmap : (x : Ordinal ) → def X x → Ordinal |
| 34 ymap : (y : Ordinal ) → def Y y → Ordinal | |
| 35 ymap-on-X : {y : Ordinal } → (lty : def Y y ) → def X (ymap y lty) | |
| 36 onto-iso : {y : Ordinal } → (lty : def Y y ) → xmap ( ymap y lty ) (ymap-on-X lty ) ≡ y | |
| 51 | 37 |
| 224 | 38 record Cardinal (X : OD ) : Set n where |
| 219 | 39 field |
| 224 | 40 cardinal : Ordinal |
| 219 | 41 conto : Onto (Ord cardinal) X |
| 224 | 42 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto (Ord y) X |
| 151 | 43 |
| 224 | 44 cardinal : (X : OD ) → Cardinal X |
| 45 cardinal X = record { | |
| 219 | 46 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
| 47 ; conto = onto | |
| 48 ; cmax = cmax | |
| 49 } where | |
| 224 | 50 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto (Ord x) X) ) |
| 219 | 51 cardinal-p x with p∨¬p ( Onto (Ord x) X ) |
| 52 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 53 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
| 224 | 54 onto-set : OD |
| 219 | 55 onto-set = record { def = λ x → {!!} } -- Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X } |
| 56 onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X | |
| 57 onto = record { | |
| 58 xmap = xmap | |
| 59 ; ymap = ymap | |
| 60 ; ymap-on-X = ymap-on-X | |
| 61 ; onto-iso = onto-iso | |
| 62 } where | |
| 63 -- | |
| 64 -- Ord cardinal itself has no onto map, but if we have x o< cardinal, there is one | |
| 65 -- od→ord X o< cardinal, so if we have def Y y or def X y, there is an Onto (Ord y) X | |
| 66 Y = (Ord (sup-o (λ x → proj1 (cardinal-p x)))) | |
| 224 | 67 lemma1 : (y : Ordinal ) → def Y y → Onto (Ord y) X |
| 68 lemma1 y y<Y with sup-o< {λ x → proj1 ( cardinal-p x)} {y} | |
| 219 | 69 ... | t = {!!} |
| 70 lemma2 : def Y (od→ord X) | |
| 71 lemma2 = {!!} | |
| 224 | 72 xmap : (x : Ordinal ) → def Y x → Ordinal |
| 219 | 73 xmap = {!!} |
| 224 | 74 ymap : (y : Ordinal ) → def X y → Ordinal |
| 219 | 75 ymap = {!!} |
| 224 | 76 ymap-on-X : {y : Ordinal } → (lty : def X y ) → def Y (ymap y lty) |
| 219 | 77 ymap-on-X = {!!} |
| 224 | 78 onto-iso : {y : Ordinal } → (lty : def X y ) → xmap (ymap y lty) (ymap-on-X lty ) ≡ y |
| 219 | 79 onto-iso = {!!} |
| 80 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X | |
| 224 | 81 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} |
| 82 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
| 219 | 83 lemma : proj1 (cardinal-p y) ≡ y |
| 84 lemma with p∨¬p ( Onto (Ord y) X ) | |
| 85 lemma | case1 x = refl | |
| 86 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 87 |
| 224 | 88 func : (f : Ordinal → Ordinal ) → OD |
| 89 func f = record { def = λ y → (x : Ordinal ) → y ≡ f x } | |
| 217 | 90 |
| 224 | 91 Func : OD |
| 92 Func = record { def = λ x → (f : Ordinal → Ordinal ) → x ≡ od→ord (func f) } | |
| 218 | 93 |
| 224 | 94 odmap : { x : OD } → Func ∋ x → Ordinal → OD |
| 95 odmap {f} lt x = record { def = λ y → def f y } | |
| 218 | 96 |
| 224 | 97 lemma1 : { x : OD } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal → Ordinal ) → ¬ ( x ≡ od→ord (func f) )) |
| 219 | 98 lemma1 = {!!} |
| 218 | 99 |
| 219 | 100 |
| 101 ----- | |
| 102 -- All cardinal is ℵ0, since we are working on Countable Ordinal, | |
| 103 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 104 |
| 105 | |
| 106 |