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1175
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1 {-# OPTIONS --allow-unsolved-metas #-}
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2
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431
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3 open import Level
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4 open import Ordinals
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5 module cardinal {n : Level } (O : Ordinals {n}) where
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6
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7 open import logic
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1095
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8 -- import OD
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9 import OD hiding ( _⊆_ )
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10
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431
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11 import ODC
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1095
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12 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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431
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13 open import Relation.Binary.PropositionalEquality
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1095
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14 open import Data.Nat.Properties
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431
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15 open import Data.Empty
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16 open import Relation.Nullary
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17 open import Relation.Binary
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18 open import Relation.Binary.Core
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19
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20 open inOrdinal O
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21 open OD O
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22 open OD.OD
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23 open ODAxiom odAxiom
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1218
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24 open import ZProduct O
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431
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25
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26 import OrdUtil
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27 import ODUtil
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28 open Ordinals.Ordinals O
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29 open Ordinals.IsOrdinals isOrdinal
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1300
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30 -- open Ordinals.IsNext isNext
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431
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31 open OrdUtil O
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32 open ODUtil O
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33
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1095
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34 _⊆_ : ( A B : HOD ) → Set n
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35 _⊆_ A B = {x : Ordinal } → odef A x → odef B x
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36
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431
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37
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38 open _∧_
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39 open _∨_
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40 open Bool
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41 open _==_
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42
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43 open HOD
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44
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1124
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45 record OrdBijection (A B : Ordinal ) : Set n where
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431
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46 field
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47 fun← : (x : Ordinal ) → odef (* A) x → Ordinal
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48 fun→ : (x : Ordinal ) → odef (* B) x → Ordinal
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49 funB : (x : Ordinal ) → ( lt : odef (* A) x ) → odef (* B) ( fun← x lt )
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50 funA : (x : Ordinal ) → ( lt : odef (* B) x ) → odef (* A) ( fun→ x lt )
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51 fiso← : (x : Ordinal ) → ( lt : odef (* B) x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x
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52 fiso→ : (x : Ordinal ) → ( lt : odef (* A) x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x
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1095
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53
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1124
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54 ordbij-refl : { a b : Ordinal } → a ≡ b → OrdBijection a b
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55 ordbij-refl {a} refl = record {
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56 fun← = λ x _ → x
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57 ; fun→ = λ x _ → x
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58 ; funB = λ x lt → lt
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59 ; funA = λ x lt → lt
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60 ; fiso← = λ x lt → refl
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61 ; fiso→ = λ x lt → refl
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62 }
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63
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1095
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64 open Injection
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1124
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65 open OrdBijection
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1095
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66
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67 record IsImage (a b : Ordinal) (iab : Injection a b ) (x : Ordinal ) : Set n where
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68 field
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69 ax : odef (* a) x
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70 bx : odef (* b) (i→ iab _ ax)
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71
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72 Image : { a b : Ordinal } → Injection a b → HOD
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73 Image {a} {b} iab = record { od = record { def = λ x → IsImage a b iab x } ; odmax = a ; <odmax = λ lt → ? }
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74
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75 image=a : ?
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76 image=a = ?
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431
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77
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78 _=c=_ : ( A B : HOD ) → Set n
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1124
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79 A =c= B = OrdBijection ( & A ) ( & B )
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431
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80
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1095
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81 c=→≡ : {A B : HOD} → A =c= B → (A ≡ ?) ∧ (B ≡ ?)
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82 c=→≡ = ?
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83
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84 ≡→c= : {A B : HOD} → A ≡ B → A =c= B
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85 ≡→c= = ?
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86
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1124
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87 open import BAlgebra O
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88
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89 _-_ : (a b : Ordinal ) → Ordinal
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90 a - b = & ( (* a) ＼ (* b) )
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91
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92 -→< : (a b : Ordinal ) → (a - b) o≤ a
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93 -→< a b = ?
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94
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95 Bernstein1 : {a b : Ordinal } → a o< b → Injection a b ∧ Injection b a → Injection (b - a) b ∧ Injection b (b - a)
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96 Bernstein1 = ?
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97
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98 Bernstein : {a b : Ordinal } → Injection a b → Injection b a → OrdBijection a b
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99 Bernstein {a} {b} iab iba = be00 where
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1095
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100 a⊆b : * a ⊆ * b
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1124
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101 a⊆b {x} ax = subst (λ k → odef (* b) k) be01 ( iB iab _ ax ) where
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102 be01 : i→ iab x ax ≡ x
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103 be01 = ?
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104 be02 : x ≡ i→ iba x ?
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1303
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105 be02 = inject iab ? ? ax ( iB iba _ ? ) ?
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1095
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106 b⊆a : * b ⊆ * a
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107 b⊆a bx = ?
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1124
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108 be05 : {a b : Ordinal } → a o< b → Injection a b → Injection b a → ⊥
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109 be05 {a} {b} a<b iab iba = TransFinite0 {λ x → (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥ }
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110 ind a b a<b iab iba where
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111 ind :(x : Ordinal) →
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112 ((y : Ordinal) → y o< x → (b : Ordinal) → y o< b → Injection y b → Injection b y → ⊥ ) →
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113 (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥
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114 ind x prev b x<b ixb ibx = ?
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115 be00 : OrdBijection a b
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116 be00 with trio< a b
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117 ... | tri< a ¬b ¬c = ⊥-elim ( be05 a iab iba )
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118 ... | tri≈ ¬a b ¬c = ordbij-refl b
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119 ... | tri> ¬a ¬b c = ⊥-elim ( be05 c iba iab )
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1095
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120
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431
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121 _c<_ : ( A B : HOD ) → Set n
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122 A c< B = ¬ ( Injection (& A) (& B) )
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123
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124 Card : OD
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1124
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125 Card = record { def = λ x → (a : Ordinal) → a o< x → ¬ OrdBijection a x }
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431
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126
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127 record Cardinal (a : Ordinal ) : Set (suc n) where
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128 field
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129 card : Ordinal
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1124
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130 ciso : OrdBijection a card
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131 cmax : (x : Ordinal) → card o< x → ¬ OrdBijection a x
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431
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132
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1095
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133 Cardinal∈ : { s : HOD } → { t : Ordinal } → Ord t ∋ s → s c< Ord t
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431
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134 Cardinal∈ = {!!}
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135
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136 Cardinal⊆ : { s t : HOD } → s ⊆ t → ( s c< t ) ∨ ( s =c= t )
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137 Cardinal⊆ = {!!}
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138
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139 Cantor1 : { u : HOD } → u c< Power u
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140 Cantor1 = {!!}
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141
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142 Cantor2 : { u : HOD } → ¬ ( u =c= Power u )
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143 Cantor2 = {!!}
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1095
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144
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145
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146
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147
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