Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 227:a4cdfc84f65f
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sun, 11 Aug 2019 18:37:33 +0900 |
| parents | 176ff97547b4 |
| children | 49736efc822b |
| 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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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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| 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 |
| 227 | 51 -- contra position of sup-o< |
| 52 -- | |
| 53 | |
| 54 record Sup ( ψ : Ordinal → Ordinal ) : Set n where | |
| 55 field | |
| 56 sup-x : Ordinal | |
| 57 sup-lb : {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ sup-x ) | |
| 58 | |
| 59 sup-o> : ( ψ : Ordinal → Ordinal ) → Sup ψ | |
| 60 sup-o> ψ = record { | |
| 61 sup-x = od→ord ( minimul (Ord (osuc (sup-o ψ))) lemma ) | |
| 62 ; sup-lb = λ {z} z<sψ → lemma1 z<sψ | |
| 63 } where | |
| 64 lemma0 : {x : Ordinal} → o∅ o< osuc x | |
| 65 lemma0 {x} with trio< o∅ (osuc x) | |
| 66 lemma0 {x} | tri< a ¬b ¬c = a | |
| 67 lemma0 {x} | tri≈ ¬a refl ¬c = ⊥-elim (¬x<0 <-osuc ) | |
| 68 lemma0 {x} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 69 lemma : ¬ (Ord (osuc (sup-o ψ)) == od∅) | |
| 70 lemma record { eq→ = eq→ ; eq← = eq← } = ¬x<0 {o∅} ( eq→ lemma0 ) | |
| 71 lemma1 : {z : Ordinal} → z o< sup-o ψ → z o< osuc (ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma))) | |
| 72 lemma1 {z} lt with trio< z (ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma))) | |
| 73 lemma1 {z} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 74 lemma1 {z} lt | tri≈ ¬a refl ¬c = <-osuc | |
| 75 lemma1 {z} lt | tri> ¬a ¬b c = ⊥-elim (o<> c lemma2 ) where | |
| 76 lemma2 : z o< ψ (od→ord (minimul (Ord (osuc (sup-o ψ))) lemma)) | |
| 77 lemma2 = ordtrans sup-o< ( o<-subst (x∋minimul (Ord (osuc (sup-o ψ))) lemma ) ? ?) | |
| 78 | |
| 219 | 79 ------------ |
| 80 -- | |
| 81 -- Onto map | |
| 82 -- def X x -> xmap | |
| 83 -- X ---------------------------> Y | |
| 84 -- ymap <- def Y y | |
| 85 -- | |
| 224 | 86 record Onto (X Y : OD ) : Set n where |
| 219 | 87 field |
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88 xmap : Ordinal |
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89 ymap : Ordinal |
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90 xfunc : def (Func X Y) xmap |
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91 yfunc : def (Func Y X) ymap |
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92 onto-iso : {y : Ordinal } → (lty : def Y y ) → Func→func (ord→od xmap) xfunc ( Func→func (ord→od ymap) yfunc y ) ≡ y |
| 51 | 93 |
| 224 | 94 record Cardinal (X : OD ) : Set n where |
| 219 | 95 field |
| 224 | 96 cardinal : Ordinal |
| 219 | 97 conto : Onto (Ord cardinal) X |
| 224 | 98 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto (Ord y) X |
| 151 | 99 |
| 224 | 100 cardinal : (X : OD ) → Cardinal X |
| 101 cardinal X = record { | |
| 219 | 102 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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103 ; conto = onto |
| 219 | 104 ; cmax = cmax |
| 105 } where | |
| 224 | 106 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto (Ord x) X) ) |
| 219 | 107 cardinal-p x with p∨¬p ( Onto (Ord x) X ) |
| 108 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 109 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } | |
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110 onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X |
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111 onto = {!!} |
| 219 | 112 cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X |
| 224 | 113 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))} |
| 114 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where | |
| 219 | 115 lemma : proj1 (cardinal-p y) ≡ y |
| 116 lemma with p∨¬p ( Onto (Ord y) X ) | |
| 117 lemma | case1 x = refl | |
| 118 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 119 |
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120 |
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121 ----- |
| 219 | 122 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 123 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 124 |
| 125 | |
| 126 |