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431
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1 {-# OPTIONS --allow-unsolved-metas #-}
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2 open import Level
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3 open import Ordinals
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4 module OD {n : Level } (O : Ordinals {n} ) where
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5
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6 open import zf
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7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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8 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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9 open import Data.Nat.Properties
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10 open import Data.Empty
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11 open import Relation.Nullary
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12 open import Relation.Binary hiding (_⇔_)
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13 open import Relation.Binary.Core hiding (_⇔_)
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14
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15 open import logic
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16 import OrdUtil
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17 open import nat
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18
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19 open Ordinals.Ordinals O
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20 open Ordinals.IsOrdinals isOrdinal
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21 open Ordinals.IsNext isNext
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22 open OrdUtil O
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23
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24 -- Ordinal Definable Set
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25
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26 record OD : Set (suc n ) where
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27 field
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28 def : (x : Ordinal ) → Set n
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29
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30 open OD
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31
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32 open _∧_
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33 open _∨_
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34 open Bool
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35
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36 record _==_ ( a b : OD ) : Set n where
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37 field
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38 eq→ : ∀ { x : Ordinal } → def a x → def b x
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39 eq← : ∀ { x : Ordinal } → def b x → def a x
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40
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41 ==-refl : { x : OD } → x == x
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42 ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x }
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43
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44 open _==_
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45
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46 ==-trans : { x y z : OD } → x == y → y == z → x == z
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47 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) }
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48
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49 ==-sym : { x y : OD } → x == y → y == x
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50 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
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51
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52
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53 ⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y
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54 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
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55 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
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56
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57 -- next assumptions are our axiom
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58 --
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59 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
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60 -- correspondence to the OD then the OD looks like a ZF Set.
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61 --
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62 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
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63 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
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64 --
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65 -- In classical Set Theory, HOD is used, as a subset of OD,
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66 -- HOD = { x | TC x ⊆ OD }
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67 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
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68 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
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69 --
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70 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
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71 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
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72 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
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73 -- bound on each HOD.
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74 --
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75 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
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76 -- we need explict assumption on sup.
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77 --
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78 -- ==→o≡ is necessary to prove axiom of extensionality.
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79
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80 -- Ordinals in OD , the maximum
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81 Ords : OD
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82 Ords = record { def = λ x → One }
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83
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84 record HOD : Set (suc n) where
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85 field
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86 od : OD
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87 odmax : Ordinal
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88 <odmax : {y : Ordinal} → def od y → y o< odmax
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89
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90 open HOD
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91
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92 record ODAxiom : Set (suc n) where
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93 field
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94 -- HOD is isomorphic to Ordinal (by means of Goedel number)
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95 & : HOD → Ordinal
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96 * : Ordinal → HOD
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97 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
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98 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
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99 *iso : {x : HOD } → * ( & x ) ≡ x
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100 &iso : {x : Ordinal } → & ( * x ) ≡ x
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101 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
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102 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
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103 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
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104 -- possible order restriction
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105 ho< : {x : HOD} → & x o< next (odmax x)
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106
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107
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108 postulate odAxiom : ODAxiom
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109 open ODAxiom odAxiom
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110
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111 -- odmax minimality
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112 --
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113 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
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114 -- We can calculate the minimum using sup but it is tedius.
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115 -- Only Select has non minimum odmax.
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116 -- We have the same problem on 'def' itself, but we leave it.
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117
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118 odmaxmin : Set (suc n)
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119 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
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120
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121 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
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122 ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
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123 ¬OD-order & * c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj )
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124
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125 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
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126 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
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127 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
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128 next-ord : Ordinal → Ordinal
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129 next-ord x = osuc x
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130
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131 -- Ordinal in OD ( and ZFSet ) Transitive Set
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132 Ord : ( a : Ordinal ) → HOD
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133 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
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134 lemma : {x : Ordinal} → x o< a → x o< a
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135 lemma {x} lt = lt
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136
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137 od∅ : HOD
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138 od∅ = Ord o∅
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139
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140 odef : HOD → Ordinal → Set n
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141 odef A x = def ( od A ) x
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142
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143 _∋_ : ( a x : HOD ) → Set n
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144 _∋_ a x = odef a ( & x )
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145
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146 -- _c<_ : ( x a : HOD ) → Set n
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147 -- x c< a = a ∋ x
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148
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149 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x)
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150 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
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151
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152 -- odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
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153 -- odef-subst df refl refl = df
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154
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155 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
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156 otrans x<a y<x = ordtrans y<x x<a
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157
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158 -- If we have reverse of c<→o<, everything becomes Ordinal
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159 ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x)
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160 ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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161 lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
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162 lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
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163 lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
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164 lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
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165
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166 -- avoiding lv != Zero error
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167 orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y
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168 orefl refl = refl
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169
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170 ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y
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171 ==-iso {x} {y} eq = record {
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172 eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ;
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173 eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) }
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174 where
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175 lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z
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176 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
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177
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178 =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y)
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179 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
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180
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181 ord→== : { x y : HOD } → & x ≡ & y → od x == od y
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182 ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
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183 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy)
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184 lemma ox ox refl = ==-refl
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185
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186 o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y)
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187 o≡→== {x} {.x} refl = ==-refl
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188
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556
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189 *≡*→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
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190 *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq )
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191
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192 &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
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193 &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq )
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194
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431
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195 o∅≡od∅ : * (o∅ ) ≡ od∅
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196 o∅≡od∅ = ==→o≡ lemma where
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197 lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x
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198 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
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199 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
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200 lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x
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201 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
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202 lemma : od (* o∅) == od od∅
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203 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
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204
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205 ord-od∅ : & (od∅ ) ≡ o∅
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206 ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
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207
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208 ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅
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209 ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt))))
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210 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
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211
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212 =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅
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213 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
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214
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448
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215 ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅
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216 ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) )
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217
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431
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218 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
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219 eq→ ∅0 {w} (lift ())
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220 eq← ∅0 {w} lt = lift (¬x<0 lt)
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221
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222 ∅< : { x y : HOD } → odef x (& y ) → ¬ ( od x == od od∅ )
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223 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
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224 ∅< {x} {y} d eq | lift ()
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450
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225
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431
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226 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
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227 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
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228
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229 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
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230 odef-iso refl t = t
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231
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232 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
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233 is-o∅ x with trio< x o∅
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234 is-o∅ x | tri< a ¬b ¬c = no ¬b
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235 is-o∅ x | tri≈ ¬a b ¬c = yes b
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236 is-o∅ x | tri> ¬a ¬b c = no ¬b
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237
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238 -- the pair
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239 _,_ : HOD → HOD → HOD
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240 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where
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241 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
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242 lemma {t} (case1 refl) = omax-x _ _
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243 lemma {t} (case2 refl) = omax-y _ _
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244
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245 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
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246 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
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247 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
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248 lemma (case1 refl) = y∋x
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249 lemma (case2 refl) = y∋x
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250
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251 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger.
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252 odmax<& : { x y : HOD } → x ∋ y → Set n
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253 odmax<& {x} {y} x∋y = odmax x o< & x
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254
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255 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
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256 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) }
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257
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258 _∩_ : ( A B : HOD ) → HOD
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259 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
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260 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
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261
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262 record _⊆_ ( A B : HOD ) : Set (suc n) where
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263 field
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264 incl : { x : HOD } → A ∋ x → B ∋ x
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265
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266 open _⊆_
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267 infixr 220 _⊆_
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268
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269 -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o<
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270 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
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271 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
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272 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
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273 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
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274 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
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275 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x )))
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276 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
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277 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
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278 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
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279 lemma (case1 refl) = refl
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280 lemma (case2 refl) = refl
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281 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
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282 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
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283 lemma1 : osuc (& y) o< & (x , x)
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284 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
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285
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286 ε-induction : { ψ : HOD → Set (suc n)}
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287 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
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288 → (x : HOD ) → ψ x
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289 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
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290 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
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291 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
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292 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
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293 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
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294
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295 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
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296 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
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297
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298 Replace : HOD → (HOD → HOD) → HOD
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299 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y)))) ∧ def (in-codomain X ψ) x }
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300 ; odmax = rmax ; <odmax = rmax<} where
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301 rmax : Ordinal
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302 rmax = sup-o X (λ y X∋y → & (ψ (* y)))
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303 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
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304 rmax< lt = proj1 lt
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305
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306 --
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307 -- If we have LEM, Replace' is equivalent to Replace
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308 --
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309 in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD
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310 in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ & (ψ (* y ) (d→∋ X lt) )))) }
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311 Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD
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312 Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x }
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313 ; odmax = rmax ; <odmax = rmax< } where
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314 rmax : Ordinal
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315 rmax = sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y)))
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316 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax
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317 rmax< lt = proj1 lt
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318
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319 Union : HOD → HOD
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320 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x))) }
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321 ; odmax = osuc (& U) ; <odmax = umax< } where
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322 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (* u)) y) → y o< osuc (& U)
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323 umax< {y} not = lemma (FExists _ lemma1 not ) where
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324 lemma0 : {x : Ordinal} → def (od (* x)) y → y o< x
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325 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) &iso &iso (c<→o< (d→∋ (* x) x<y ))
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326 lemma2 : {x : Ordinal} → def (od U) x → x o< & U
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327 lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U))
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328 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (* x)) y → ¬ (& U o< y)
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329 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
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330 lemma : ¬ ((& U) o< y ) → y o< osuc (& U)
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331 lemma not with trio< y (& U)
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332 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
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333 lemma not | tri≈ ¬a refl ¬c = <-osuc
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334 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
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335 _∈_ : ( A B : HOD ) → Set n
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336 A ∈ B = B ∋ A
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337
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338 OPwr : (A : HOD ) → HOD
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339 OPwr A = Ord ( sup-o (Ord (osuc (& A))) ( λ x A∋x → & ( A ∩ (* x)) ) )
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340
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341 Power : HOD → HOD
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342 Power A = Replace (OPwr (Ord (& A))) ( λ x → A ∩ x )
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343 -- ｛_｝ : ZFSet → ZFSet
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344 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly
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345
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346 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
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347 union→ X z u xx not = ⊥-elim ( not (& u) ( ⟪ proj1 xx
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348 , subst ( λ k → odef k (& z)) (sym *iso) (proj2 xx) ⟫ ))
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349 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
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350 union← X z UX∋z = FExists _ lemma UX∋z where
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351 lemma : {y : Ordinal} → odef X y ∧ odef (* y) (& z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
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352 lemma {y} xx not = not (* y) ⟪ d→∋ X (proj1 xx) , proj2 xx ⟫
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353
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354 data infinite-d : ( x : Ordinal ) → Set n where
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355 iφ : infinite-d o∅
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356 isuc : {x : Ordinal } → infinite-d x →
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357 infinite-d (& ( Union (* x , (* x , * x ) ) ))
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358
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359 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
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360 -- We simply assumes infinite-d y has a maximum.
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361 --
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362 -- This means that many of OD may not be HODs because of the & mapping divergence.
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363 -- We should have some axioms to prevent this such as & x o< next (odmax x).
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364 --
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365 -- postulate
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366 -- ωmax : Ordinal
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367 -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
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368 --
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369 -- infinite : HOD
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370 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
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371
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372 infinite : HOD
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373 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
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374 u : (y : Ordinal ) → HOD
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375 u y = Union (* y , (* y , * y))
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376 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
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377 lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y))
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378 lemma8 = ho<
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379 --- (x,y) < next (omax x y) < next (osuc y) = next y
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380 lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y)
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381 lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
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382 lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y))
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383 lemma81 {y} = nexto=n (subst (λ k → & (* y , * y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
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384 lemma9 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y , * y))
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385 lemma9 = lemmaa (c<→o< (case1 refl))
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386 lemma71 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y))
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387 lemma71 = next< lemma81 lemma9
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388 lemma1 : {y : Ordinal} → & (u y) o< next (osuc (& (* y , (* y , * y))))
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389 lemma1 = ho<
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390 --- main recursion
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391 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
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392 lemma {o∅} iφ = x<nx
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393 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1))
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394
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395 empty : (x : HOD ) → ¬ (od∅ ∋ x)
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396 empty x = ¬x<0
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397
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398 _=h=_ : (x y : HOD) → Set n
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399 x =h= y = od x == od y
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400
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401 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
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402 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
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403 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
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404
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405 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
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406 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
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407 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
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408
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409 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
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410 o<→c< lt = record { incl = λ z → ordtrans z lt }
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411
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412 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
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413 ⊆→o< {x} {y} lt with trio< x y
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414 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
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415 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
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416 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym &iso) refl )
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417 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
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418
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419 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
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420 ψiso {ψ} t refl = t
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421 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
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422 selection {ψ} {X} {y} = ⟪
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423 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ )
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424 , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ )
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425 ⟫
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426
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427 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
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428 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
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429
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430 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → & (ψ x) o< (sup-o X (λ y X∋y → & (ψ (* y))))
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431 sup-c< ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o< X lt )
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432 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
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433 replacement← {ψ} X x lt = ⟪ sup-c< ψ {X} {x} lt , lemma ⟫ where
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434 lemma : def (in-codomain X ψ) (& (ψ x))
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435 lemma not = ⊥-elim ( not ( & x ) ⟪ lt , cong (λ k → & (ψ k)) (sym *iso)⟫ )
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436 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
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437 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
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438 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((& x) ≡ & (ψ (* y))))
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439 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)))
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440 lemma2 not not2 = not ( λ y d → not2 y ⟪ proj1 d , lemma3 (proj2 d)⟫) where
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441 lemma3 : {y : Ordinal } → (& x ≡ & (ψ (* y))) → (* (& x) =h= ψ (* y))
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442 lemma3 {y} eq = subst (λ k → * (& x) =h= k ) *iso (o≡→== eq )
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443 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)) )
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444 lemma not y not2 = not (* y) (subst (λ k → k =h= ψ (* y)) *iso ( proj2 not2 ))
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445
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446 ---
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447 --- Power Set
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448 ---
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449 --- First consider ordinals in HOD
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450 ---
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451 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
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452 --
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453 --
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454 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
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455 ∩-≡ {a} {b} inc = record {
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456 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ;
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457 eq← = λ {x} x<a∩b → proj2 x<a∩b }
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458 --
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459 -- Transitive Set case
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460 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
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461 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
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462 -- OPwr A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) )
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463 --
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464 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
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465 ord-power← a t t→A = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where
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466 lemma-eq : ((Ord a) ∩ t) =h= t
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467 eq→ lemma-eq {z} w = proj2 w
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468 eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫
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469 lemma1 : {a : Ordinal } { t : HOD }
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470 → (eq : ((Ord a) ∩ t) =h= t) → & ((Ord a) ∩ (* (& t))) ≡ & t
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471 lemma1 {a} {t} eq = subst (λ k → & ((Ord a) ∩ k) ≡ & t ) (sym *iso) (cong (λ k → & k ) (==→o≡ eq ))
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472 lemma2 : (& t) o< (osuc (& (Ord a)))
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473 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t)))
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474 lemma : & ((Ord a) ∩ (* (& t)) ) o< sup-o (Ord (osuc (& (Ord a)))) (λ x lt → & ((Ord a) ∩ (* x)))
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475 lemma = sup-o< _ lemma2
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476
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477 --
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478 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (& A)) first
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479 -- then replace of all elements of the Power set by A ∩ y
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480 --
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481 -- Power A = Replace (OPwr (Ord (& A))) ( λ y → A ∩ y )
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482
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483 -- we have oly double negation form because of the replacement axiom
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484 --
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485 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
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486 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
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487 a = & A
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488 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
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489 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (& A))) t P∋t
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490 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
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491 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
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492 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ * y)))
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493 lemma4 not = lemma2 ( λ y not1 → not (& y) (subst (λ k → t =h= ( A ∩ k )) (sym *iso) not1 ))
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494 lemma5 : {y : Ordinal} → t =h= (A ∩ * y) → ¬ ¬ (odef A (& x))
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495 lemma5 {y} eq not = (lemma3 (* y) eq) not
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496
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497 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
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498 power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where
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499 a = & A
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500 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
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501 lemma0 {x} t∋x = c<→o< (t→A t∋x)
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502 lemma3 : OPwr (Ord a) ∋ t
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503 lemma3 = ord-power← a t lemma0
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504 lemma4 : (A ∩ * (& t)) ≡ t
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505 lemma4 = let open ≡-Reasoning in begin
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506 A ∩ * (& t)
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507 ≡⟨ cong (λ k → A ∩ k) *iso ⟩
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508 A ∩ t
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509 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
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510 t
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511 ∎
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512 sup1 : Ordinal
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513 sup1 = sup-o (Ord (osuc (& (Ord (& A))))) (λ x A∋x → & ((Ord (& A)) ∩ (* x)))
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514 lemma9 : def (od (Ord (Ordinals.osuc O (& (Ord (& A)))))) (& (Ord (& A)))
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515 lemma9 = <-osuc
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516 lemmab : & ((Ord (& A)) ∩ (* (& (Ord (& A) )))) o< sup1
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517 lemmab = sup-o< (Ord (osuc (& (Ord (& A))))) lemma9
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518 lemmad : Ord (osuc (& A)) ∋ t
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519 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt)))
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520 lemmac : ((Ord (& A)) ∩ (* (& (Ord (& A) )))) =h= Ord (& A)
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521 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
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522 lemmaf : {x : Ordinal} → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x → def (od (Ord (& A))) x
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523 lemmaf {x} lt = proj1 lt
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524 lemmag : {x : Ordinal} → def (od (Ord (& A))) x → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x
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525 lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫
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526 lemmae : & ((Ord (& A)) ∩ (* (& (Ord (& A))))) ≡ & (Ord (& A))
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527 lemmae = cong (λ k → & k ) ( ==→o≡ lemmac)
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528 lemma7 : def (od (OPwr (Ord (& A)))) (& t)
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529 lemma7 with osuc-≡< lemmad
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530 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
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531 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {* (& t)} {* (& (Ord (& t)))} (λ {x} lt → lemmah lt )) where
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532 lemmah : {x : Ordinal } → def (od (* (& t))) x → def (od (* (& (Ord (& t))))) x
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533 lemmah {x} lt = subst (λ k → def (od k) x ) (sym *iso) (subst (λ k → k o< (& t))
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534 &iso
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535 (c<→o< (subst₂ (λ j k → def (od j) k) *iso (sym &iso) lt )))
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536 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
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537 lemmai : & (Ord (& A)) ≡ & t
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538 lemmai = let open ≡-Reasoning in begin
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539 & (Ord (& A))
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540 ≡⟨ sym (cong (λ k → & (Ord k)) eq) ⟩
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541 & (Ord (& t))
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542 ≡⟨ sym &iso ⟩
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543 & (* (& (Ord (& t))))
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544 ≡⟨ sym eq1 ⟩
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545 & (* (& t))
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546 ≡⟨ &iso ⟩
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547 & t
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548 ∎
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549 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
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550 lemmak : & (* (& (Ord (& t)))) ≡ & (Ord (& A))
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551 lemmak = let open ≡-Reasoning in begin
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552 & (* (& (Ord (& t))))
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553 ≡⟨ &iso ⟩
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554 & (Ord (& t))
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555 ≡⟨ cong (λ k → & (Ord k)) eq ⟩
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556 & (Ord (& A))
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557 ∎
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558 lemmaj : & t o< & (Ord (& A))
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559 lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt
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560 lemma1 : & t o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x)))
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561 lemma1 = subst (λ k → & k o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x))))
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562 lemma4 (sup-o< (OPwr (Ord (& A))) lemma7 )
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563 lemma2 : def (in-codomain (OPwr (Ord (& A))) (_∩_ A)) (& t)
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564 lemma2 not = ⊥-elim ( not (& t) ⟪ lemma3 , lemma6 ⟫ ) where
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565 lemma6 : & t ≡ & (A ∩ * (& t))
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566 lemma6 = cong ( λ k → & k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym *iso) ( ∩-≡ {t} {A} t→A )))
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567
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568
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569 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
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570 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d
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571 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d
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572
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573 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
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574 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
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575 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
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576
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577 infinity∅ : infinite ∋ od∅
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578 infinity∅ = subst (λ k → odef infinite k ) lemma iφ where
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579 lemma : o∅ ≡ & od∅
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580 lemma = let open ≡-Reasoning in begin
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581 o∅
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582 ≡⟨ sym &iso ⟩
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583 & ( * o∅ )
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584 ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
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585 & od∅
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586 ∎
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587 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
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588 infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where
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589 lemma : & (Union (* (& x) , (* (& x) , * (& x))))
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590 ≡ & (Union (x , (x , x)))
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591 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
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592
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593 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
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594 isZF = record {
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595 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
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596 ; pair→ = pair→
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597 ; pair← = pair←
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598 ; union→ = union→
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599 ; union← = union←
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600 ; empty = empty
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601 ; power→ = power→
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602 ; power← = power←
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603 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
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604 ; ε-induction = ε-induction
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605 ; infinity∅ = infinity∅
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606 ; infinity = infinity
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607 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
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608 ; replacement← = replacement←
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609 ; replacement→ = λ {ψ} → replacement→ {ψ}
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610 }
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611
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612 HOD→ZF : ZF
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613 HOD→ZF = record {
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614 ZFSet = HOD
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615 ; _∋_ = _∋_
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616 ; _≈_ = _=h=_
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617 ; ∅ = od∅
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618 ; _,_ = _,_
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619 ; Union = Union
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620 ; Power = Power
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621 ; Select = Select
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622 ; Replace = Replace
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623 ; infinite = infinite
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624 ; isZF = isZF
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625 }
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626
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627
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