Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 226:176ff97547b4
set theortic function definition using sup
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 11 Aug 2019 13:05:17 +0900 |
| parents | 5f48299929ac |
| children | a4cdfc84f65f |
| 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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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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fce60b99dc55
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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fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
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23 |
| 225 | 24 |
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25 func : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
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26 func f dom = Replace dom ( λ x → x , (ord→od (f (od→ord x) ))) |
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27 |
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28 record _⊗_ (A B : Ordinal) : Set n where |
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29 field |
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30 π1 : Ordinal |
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31 π2 : Ordinal |
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32 A∋π1 : def (ord→od A) π1 |
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33 B∋π2 : def (ord→od B) π2 |
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34 |
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35 Func : ( A B : OD ) → OD |
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36 Func A B = record { def = λ x → (od→ord A) ⊗ (od→ord B) } |
| 225 | 37 |
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38 π1 : { A B x : OD } → Func A B ∋ x → OD |
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39 π1 {A} {B} {x} p = ord→od (_⊗_.π1 p) |
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40 |
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41 π2 : { A B x : OD } → Func A B ∋ x → OD |
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42 π2 {A} {B} {x} p = ord→od (_⊗_.π2 p) |
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43 |
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44 Func→func : { dom cod : OD } → (f : OD ) → Func dom cod ∋ f → (Ordinal → Ordinal ) |
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45 Func→func {dom} {cod} f lt x = sup-o ( λ y → lemma y ) where |
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46 lemma : Ordinal → Ordinal |
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47 lemma y with p∨¬p ( _⊗_.π1 lt ≡ x ) |
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48 lemma y | case1 refl = _⊗_.π2 lt |
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49 lemma y | case2 not = o∅ |
| 225 | 50 |
| 219 | 51 ------------ |
| 52 -- | |
| 53 -- Onto map | |
| 54 -- def X x -> xmap | |
| 55 -- X ---------------------------> Y | |
| 56 -- ymap <- def Y y | |
| 57 -- | |
| 224 | 58 record Onto (X Y : OD ) : Set n where |
| 219 | 59 field |
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60 xmap : Ordinal |
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61 ymap : Ordinal |
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62 xfunc : def (Func X Y) xmap |
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63 yfunc : def (Func Y X) ymap |
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64 onto-iso : {y : Ordinal } → (lty : def Y y ) → Func→func (ord→od xmap) xfunc ( Func→func (ord→od ymap) yfunc y ) ≡ y |
| 51 | 65 |
| 224 | 66 record Cardinal (X : OD ) : Set n where |
| 219 | 67 field |
| 224 | 68 cardinal : Ordinal |
| 219 | 69 conto : Onto (Ord cardinal) X |
| 224 | 70 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto (Ord y) X |
| 151 | 71 |
| 224 | 72 cardinal : (X : OD ) → Cardinal X |
| 73 cardinal X = record { | |
| 219 | 74 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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75 ; conto = onto |
| 219 | 76 ; cmax = cmax |
| 77 } where | |
| 224 | 78 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto (Ord x) X) ) |
| 219 | 79 cardinal-p x with p∨¬p ( Onto (Ord x) X ) |
| 80 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 81 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
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82 onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X |
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83 onto = {!!} |
| 219 | 84 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X |
| 224 | 85 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} |
| 86 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
| 219 | 87 lemma : proj1 (cardinal-p y) ≡ y |
| 88 lemma with p∨¬p ( Onto (Ord y) X ) | |
| 89 lemma | case1 x = refl | |
| 90 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 91 |
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92 |
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93 ----- |
| 219 | 94 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 95 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 96 |
| 97 | |
| 98 |