Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 225:5f48299929ac
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 11 Aug 2019 08:10:13 +0900 |
| parents | afc864169325 |
| children | 176ff97547b4 |
| rev | line source |
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| 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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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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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 |
| 225 | 24 |
| 25 func : (f : Ordinal → Ordinal ) → ( dom cod : OD ) → OD | |
| 26 func f dom cod = record { def = λ z → {x y : Ordinal} → (z ≡ omax x y ) ∧ def dom x ∧ def cod (f x ) } | |
| 27 | |
| 28 -- Func : ( dom cod : OD ) → OD | |
| 29 -- Func dom cod = record { def = λ x → x o< sup-o ( λ y → (f : Ordinal → Ordinal ) → y ≡ od→ord (func f dom cod) ) } | |
| 30 | |
| 219 | 31 ------------ |
| 32 -- | |
| 33 -- Onto map | |
| 34 -- def X x -> xmap | |
| 35 -- X ---------------------------> Y | |
| 36 -- ymap <- def Y y | |
| 37 -- | |
| 224 | 38 record Onto (X Y : OD ) : Set n where |
| 219 | 39 field |
| 224 | 40 xmap : (x : Ordinal ) → def X x → Ordinal |
| 41 ymap : (y : Ordinal ) → def Y y → Ordinal | |
| 42 ymap-on-X : {y : Ordinal } → (lty : def Y y ) → def X (ymap y lty) | |
| 43 onto-iso : {y : Ordinal } → (lty : def Y y ) → xmap ( ymap y lty ) (ymap-on-X lty ) ≡ y | |
| 51 | 44 |
| 224 | 45 record Cardinal (X : OD ) : Set n where |
| 219 | 46 field |
| 224 | 47 cardinal : Ordinal |
| 219 | 48 conto : Onto (Ord cardinal) X |
| 224 | 49 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto (Ord y) X |
| 151 | 50 |
| 224 | 51 cardinal : (X : OD ) → Cardinal X |
| 52 cardinal X = record { | |
| 219 | 53 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
| 225 | 54 ; conto = x∋minimul onto-set ∃-onto-set |
| 219 | 55 ; cmax = cmax |
| 56 } where | |
| 224 | 57 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto (Ord x) X) ) |
| 219 | 58 cardinal-p x with p∨¬p ( Onto (Ord x) X ) |
| 59 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 60 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
| 224 | 61 onto-set : OD |
| 225 | 62 onto-set = record { def = λ x → Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X } |
| 63 ∃-onto-set : ¬ ( onto-set == od∅ ) | |
| 64 ∃-onto-set record { eq→ = eq→ ; eq← = eq← } = ¬x<0 {_} ( eq→ lemma ) where | |
| 65 lemma : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X | |
| 66 lemma = {!!} | |
| 219 | 67 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X |
| 224 | 68 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} |
| 69 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
| 219 | 70 lemma : proj1 (cardinal-p y) ≡ y |
| 71 lemma with p∨¬p ( Onto (Ord y) X ) | |
| 72 lemma | case1 x = refl | |
| 73 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 74 |
| 219 | 75 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 76 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 77 |
| 78 | |
| 79 |