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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 Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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7 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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8 open import Data.Nat.Properties
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9 open import Data.Empty
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10 open import Data.Unit
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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 --
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58 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
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59 -- correspondence to the OD then the OD looks like a ZF Set.
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60 --
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61 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
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62 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
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63 --
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64 -- In classical Set Theory, HOD is used, as a subset of OD,
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65 -- HOD = { x | TC x ⊆ OD }
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66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
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67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
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68 --
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69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
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70 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
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71 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
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72 -- bound on each HOD.
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73 --
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74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
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75 -- we need explict assumption on sup for unrestricted Replacement.
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76 --
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77 -- ==→o≡ is necessary to prove axiom of extensionality.
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78
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79 -- Ordinals in OD , the maximum
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80 Ords : OD
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81 Ords = record { def = λ x → Lift n ⊤ }
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82
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83 record HOD : Set (suc n) where
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84 field
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85 od : OD
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86 odmax : Ordinal
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87 <odmax : {y : Ordinal} → def od y → y o< odmax
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88
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89 open HOD
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90
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91 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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92
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93 record ODAxiom : Set (suc n) where
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94 field
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95 -- HOD is isomorphic to Ordinal (by means of Goedel number)
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96 & : HOD → Ordinal
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97 * : Ordinal → HOD
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98 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
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99 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
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100 *iso : {x : HOD } → * ( & x ) ≡ x
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101 &iso : {x : Ordinal } → & ( * x ) ≡ x
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102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
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103 ∋-irr : {X : HOD} {x : Ordinal } → (a b : def (od X) x) → a ≅ b
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104
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105 postulate odAxiom : ODAxiom
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106 open ODAxiom odAxiom
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107
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108 -- possible order restriction (required in the axiom of Omega )
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109
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110 -- postulate odAxiom-ho< : ODAxiom-ho<
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111 -- open ODAxiom-ho< odAxiom-ho<
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112
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113 -- odmax minimality
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114 --
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115 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
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116 -- We can calculate the minimum using sup but it is tedius.
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117 -- Only Select has non minimum odmax.
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118 -- We have the same problem on 'def' itself, but we leave it.
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119
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120 odmaxmin : Set (suc n)
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121 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
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122
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123 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
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124 ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
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125 ¬OD-order & * c<→o< = o≤> <-osuc (c<→o< {Ords} (lift tt) )
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126
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127 -- Ordinal in OD ( and ZFSet ) Transitive Set
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128 Ord : ( a : Ordinal ) → HOD
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129 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
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130 lemma : {x : Ordinal} → x o< a → x o< a
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131 lemma {x} lt = lt
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132
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133 od∅ : HOD
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134 od∅ = Ord o∅
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135
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136 odef : HOD → Ordinal → Set n
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137 odef A x = def ( od A ) x
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138
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139 _∋_ : ( a x : HOD ) → Set n
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140 _∋_ a x = odef a ( & x )
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141
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142 -- _c<_ : ( x a : HOD ) → Set n
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143 -- x c< a = a ∋ x
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144
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145 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x)
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146 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
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147
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148 -- odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
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149 -- odef-subst df refl refl = df
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150
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151 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
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152 otrans x<a y<x = ordtrans y<x x<a
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153
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154 -- If we have reverse of c<→o<, everything becomes Ordinal
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155 ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x)
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156 ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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157 lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
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158 lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
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159 lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
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160 lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
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161
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162 -- avoiding lv != Zero error
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163 orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y
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164 orefl refl = refl
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165
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166 ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y
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167 ==-iso {x} {y} eq = record {
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168 eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ;
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169 eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) }
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170 where
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171 lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z
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172 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
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173
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174 =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y)
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175 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
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176
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177 ord→== : { x y : HOD } → & x ≡ & y → od x == od y
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178 ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
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179 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy)
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180 lemma ox ox refl = ==-refl
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181
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182 o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y)
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183 o≡→== {x} {.x} refl = ==-refl
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184
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185 *≡*→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
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186 *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq )
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187
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188 &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
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189 &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq )
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190
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191 o∅≡od∅ : * (o∅ ) ≡ od∅
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192 o∅≡od∅ = ==→o≡ lemma where
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193 lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x
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194 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
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195 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
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196 lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x
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197 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
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198 lemma : od (* o∅) == od od∅
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199 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
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200
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201 ord-od∅ : & (od∅ ) ≡ o∅
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202 ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
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203
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204 ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅
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205 ≡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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206 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
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207
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208 =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅
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209 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
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210
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211 ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅
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212 ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) )
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213
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214 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
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215 eq→ ∅0 {w} (lift ())
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216 eq← ∅0 {w} lt = lift (¬x<0 lt)
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217
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218 ∅< : { x y : HOD } → odef x (& y ) → ¬ ( od x == od od∅ )
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219 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
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220 ∅< {x} {y} d eq | lift ()
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221
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222 ¬x∋y→x≡od∅ : { x : HOD } → ({y : Ordinal } → ¬ odef x y ) → x ≡ od∅
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223 ¬x∋y→x≡od∅ {x} nxy = ==→o≡ record { eq→ = λ {y} lt → ⊥-elim (nxy lt) ; eq← = λ {y} lt → ⊥-elim (¬x<0 lt) }
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224
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225 0<P→ne : { x : HOD } → o∅ o< & x → ¬ ( od x == od od∅ )
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226 0<P→ne {x} 0<x eq = ⊥-elim ( o<¬≡ (sym (=od∅→≡o∅ eq)) 0<x )
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227
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228 ∈∅< : { x : HOD } {y : Ordinal } → odef x y → o∅ o< (& x)
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229 ∈∅< {x} {y} d with trio< o∅ (& x)
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230 ... | tri< a ¬b ¬c = a
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231 ... | tri≈ ¬a b ¬c = ⊥-elim ( ∅< {x} {* y} (subst (λ k → odef x k ) (sym &iso) d ) ( ≡o∅→=od∅ (sym b) ) )
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232 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
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233
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234 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
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235 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
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236
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237 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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238 odef-iso refl t = t
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239
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240 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
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241 is-o∅ x with trio< x o∅
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242 is-o∅ x | tri< a ¬b ¬c = no ¬b
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243 is-o∅ x | tri≈ ¬a b ¬c = yes b
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244 is-o∅ x | tri> ¬a ¬b c = no ¬b
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245
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246 odef< : {b : Ordinal } { A : HOD } → odef A b → b o< & A
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247 odef< {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
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248
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249 odef∧< : {A : HOD } {y : Ordinal} {n : Level } → {P : Set n} → odef A y ∧ P → y o< & A
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250 odef∧< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
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251
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252 -- the pair
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253 _,_ : HOD → HOD → HOD
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254 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where
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255 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
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256 lemma {t} (case1 refl) = omax-x _ _
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257 lemma {t} (case2 refl) = omax-y _ _
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258
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259 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
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260 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
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261 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
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262 lemma (case1 refl) = y∋x
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263 lemma (case2 refl) = y∋x
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264
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265 -- another possible restriction. We require no minimality on odmax, so it may arbitrary larger.
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266 odmax<& : { x y : HOD } → x ∋ y → Set n
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267 odmax<& {x} {y} x∋y = odmax x o< & x
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268
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269 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
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270 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) }
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271
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272 _∩_ : ( A B : HOD ) → HOD
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273 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
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274 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
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275
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276 _⊆_ : ( A B : HOD) → Set n
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277 _⊆_ A B = { x : Ordinal } → odef A x → odef B x
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278
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279 infixr 220 _⊆_
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280
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281 -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o<
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282 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
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283 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
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1091
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284 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
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431
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285 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
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286 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
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287 ⊆→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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288 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
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289 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
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290 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
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291 lemma (case1 refl) = refl
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292 lemma (case2 refl) = refl
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293 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
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1091
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294 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
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431
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295 lemma1 : osuc (& y) o< & (x , x)
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1091
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296 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
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431
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297
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298 ε-induction : { ψ : HOD → Set (suc n)}
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299 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
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300 → (x : HOD ) → ψ x
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301 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
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302 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
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1091
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303 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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431
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304 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
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305 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
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306
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1109
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307 ε-induction0 : { ψ : HOD → Set n}
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308 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
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309 → (x : HOD ) → ψ x
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310 ε-induction0 {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
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311 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
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312 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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313 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
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314 ε-induction-ord ox {oy} lt = inOrdinal.TransFinite0 O {λ oy → ψ (* oy)} induction oy
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315
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1091
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316 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
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317 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
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318 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
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319 next-ord : Ordinal → Ordinal
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320 next-ord x = osuc x
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321
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322 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
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431
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323 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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324
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1095
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325 _=h=_ : (x y : HOD) → Set n
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326 x =h= y = od x == od y
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327
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328 record Own (A : HOD) (x : Ordinal) : Set n where
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329 field
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330 owner : Ordinal
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331 ao : odef A owner
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332 ox : odef (* owner) x
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333
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334 Union : HOD → HOD
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335 Union U = record { od = record { def = λ x → Own U x } ; odmax = osuc (& U) ; <odmax = umax } where
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336 umax : {y : Ordinal} → Own U y → y o< osuc (& U)
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337 umax {y} uy = o<→≤ ( ordtrans (odef< (Own.ox uy)) (subst (λ k → k o< & U) (sym &iso) umax1) ) where
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338 umax1 : Own.owner uy o< & U
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339 umax1 = odef< (Own.ao uy)
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340
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341 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
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342 union→ X z u xx = record { owner = & u ; ao = proj1 xx ; ox = subst (λ k → odef k (& z)) (sym *iso) (proj2 xx) }
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343 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
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344 union← X z UX∋z not = ⊥-elim ( not (* (Own.owner UX∋z)) ⟪ subst (λ k → odef X k) (sym &iso) ( Own.ao UX∋z) , Own.ox UX∋z ⟫ )
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345
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1303
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346 --
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347 --
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348 --
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349
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1285
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350 record RCod (COD : HOD) (ψ : HOD → HOD) : Set (suc n) where
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351 field
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352 ≤COD : ∀ {x : HOD } → ψ x ⊆ COD
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353
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1095
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354 record Replaced (A : HOD) (ψ : Ordinal → Ordinal ) (x : Ordinal ) : Set n where
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355 field
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356 z : Ordinal
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357 az : odef A z
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358 x=ψz : x ≡ ψ z
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359
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1285
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360 Replace : (D : HOD) → (ψ : HOD → HOD) → {C : HOD} → RCod C ψ → HOD
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361 Replace X ψ {C} rc = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = osuc (& C)
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362 ; <odmax = rmax< } where
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363 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< osuc (& C)
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364 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RCod.≤COD rc) ) where
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1095
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365 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
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366 r01 = sym (Replaced.x=ψz lt )
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367
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1285
|
368 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → {C : HOD} → (rc : RCod C ψ) → Replace X ψ rc ∋ ψ x
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369 replacement← {ψ} X x lt {C} rc = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
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370 replacement→ : {ψ : HOD → HOD} (X x : HOD) → {C : HOD} → (rc : RCod C ψ ) → (lt : Replace X ψ rc ∋ x)
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371 → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
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372 replacement→ {ψ} X x {C} rc lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
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431
|
373
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374 --
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1091
|
375 -- If we have LEM, Replace' is equivalent to Replace
|
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431
|
376 --
|
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1095
|
377
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1285
|
378 record RXCod (X COD : HOD) (ψ : (x : HOD) → X ∋ x → HOD) : Set (suc n) where
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379 field
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380 ≤COD : ∀ {x : HOD } → (lt : X ∋ x) → ψ x lt ⊆ COD
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381
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1095
|
382 record Replaced1 (A : HOD) (ψ : (x : Ordinal ) → odef A x → Ordinal ) (x : Ordinal ) : Set n where
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383 field
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384 z : Ordinal
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385 az : odef A z
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386 x=ψz : x ≡ ψ z az
|
|
431
|
387
|
|
1285
|
388 Replace' : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → RXCod X C ψ → HOD
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389 Replace' X ψ {C} rc = record { od = record { def = λ x → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) x } ; odmax = osuc (& C) ; <odmax = rmax< } where
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390 rmax< : {y : Ordinal} → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) y → y o< osuc (& C)
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391 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RXCod.≤COD rc (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) ))) where
|
|
1095
|
392 r01 : & (ψ ( * (Replaced1.z lt ) ) (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) )) ≡ y
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393 r01 = sym (Replaced1.x=ψz lt )
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394
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|
1285
|
395 cod-conv : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
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396 → RXCod (* (& X)) C (λ y xy → ψ y (subst (λ k → k ∋ y) *iso xy))
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397 cod-conv X ψ {C} rc = record { ≤COD = λ {x} lt → RXCod.≤COD rc (subst (λ k → odef k (& x)) *iso lt) }
|
|
1218
|
398
|
|
1294
|
399 Replace'-iso : {X Y : HOD} → {fx : (x : HOD) → X ∋ x → HOD} {fy : (x : HOD) → Y ∋ x → HOD}
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400 → {CX : HOD} → (rcx : RXCod X CX fx ) → {CY : HOD} → (rcy : RXCod Y CY fy )
|
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|
401 → X ≡ Y → ( (x : HOD) → (xx : X ∋ x ) → (yy : Y ∋ x ) → fx _ xx ≡ fy _ yy )
|
|
|
402 → Replace' X fx rcx ≡ Replace' Y fy rcy
|
|
|
403 Replace'-iso {X} {X} {fx} {fy} _ _ refl eq = ==→o≡ record { eq→ = ri0 ; eq← = ri1 } where
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|
|
404 ri0 : {x : Ordinal} → Replaced1 X (λ z xz → & (fx (* z) (subst (odef X) (sym &iso) xz))) x
|
|
|
405 → Replaced1 X (λ z xz → & (fy (* z) (subst (odef X) (sym &iso) xz))) x
|
|
|
406 ri0 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = az ; x=ψz = trans x=ψz (cong (&) ( eq _ xz xz )) } where
|
|
|
407 xz : X ∋ * z
|
|
|
408 xz = subst (λ k → odef X k ) (sym &iso) az
|
|
|
409 ri1 : {x : Ordinal} → Replaced1 X (λ z xz → & (fy (* z) (subst (odef X) (sym &iso) xz))) x
|
|
|
410 → Replaced1 X (λ z xz → & (fx (* z) (subst (odef X) (sym &iso) xz))) x
|
|
|
411 ri1 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = az ; x=ψz = trans x=ψz (cong (&) (sym ( eq _ xz xz ))) } where
|
|
|
412 xz : X ∋ * z
|
|
|
413 xz = subst (λ k → odef X k ) (sym &iso) az
|
|
|
414
|
|
|
415 Replace'-iso1 : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
|
|
1285
|
416 → Replace' (* (& X)) (λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) ) (cod-conv X ψ rc)
|
|
|
417 ≡ Replace' X ( λ y xy → ψ y xy ) rc
|
|
1294
|
418 Replace'-iso1 X ψ rc = Replace'-iso {* (& X)} {X} {λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) } { λ y xy → ψ y xy }
|
|
|
419 (cod-conv X ψ rc) rc
|
|
|
420 *iso (λ x xx yx → fi00 x xx yx ) where
|
|
|
421 fi00 : (x : HOD ) → (xx : (* (& X)) ∋ x ) → (yx : X ∋ x) → ψ x (subst (λ k → k ∋ x) *iso xx) ≡ ψ x yx
|
|
|
422 fi00 x xx yx = cong (λ k → ψ x k ) ( HE.≅-to-≡ ( ∋-irr {X} {& x} (subst (λ k → k ∋ x) *iso xx) yx ) )
|
|
1218
|
423
|
|
1095
|
424 -- replacement←1 : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace1 X ψ ∋ ψ x
|
|
|
425 -- replacement←1 {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
|
|
|
426 -- replacement→1 : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace1 X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
|
|
|
427 -- replacement→1 {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
|
|
|
428
|
|
431
|
429 _∈_ : ( A B : HOD ) → Set n
|
|
|
430 A ∈ B = B ∋ A
|
|
|
431
|
|
1095
|
432 Power : HOD → HOD
|
|
1453
|
433 Power A = record { od = record { def = λ x → ( z : Ordinal) → odef (* x) z → odef A z } ; odmax = osuc (& A)
|
|
1095
|
434 ; <odmax = p00 } where
|
|
|
435 p00 : {y : Ordinal} → ((z : Ordinal) → odef (* y) z → odef A z) → y o< osuc (& A)
|
|
|
436 p00 {y} y⊆A = p01 where
|
|
|
437 p01 : y o≤ & A
|
|
|
438 p01 = subst (λ k → k o≤ & A) &iso ( ⊆→o≤ (λ {x} yx → y⊆A x yx ))
|
|
431
|
439
|
|
1095
|
440 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
|
|
|
441 power→ A t P∋t {x} t∋x = P∋t (& x) (subst (λ k → odef k (& x) ) (sym *iso) t∋x )
|
|
|
442
|
|
|
443 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
444 power← A t t⊆A z xz = subst (λ k → odef A k ) &iso ( t⊆A (subst₂ (λ j k → odef j k) *iso (sym &iso) xz ))
|
|
|
445
|
|
1453
|
446 Power∋∅ : {S : HOD} → odef (Power S) o∅
|
|
|
447 Power∋∅ z xz = ⊥-elim (¬x<0 (subst (λ k → odef k z) o∅≡od∅ xz) )
|
|
|
448
|
|
1180
|
449 Intersection : (X : HOD ) → HOD -- ∩ X
|
|
1186
|
450 Intersection X = record { od = record { def = λ x → (x o≤ & X ) ∧ ( {y : Ordinal} → odef X y → odef (* y) x )} ; odmax = osuc (& X) ; <odmax = λ lt → proj1 lt }
|
|
1180
|
451
|
|
1300
|
452 empty : (x : HOD ) → ¬ (od∅ ∋ x)
|
|
|
453 empty x = ¬x<0
|
|
|
454
|
|
1180
|
455
|
|
431
|
456 -- ｛_｝ : ZFSet → ZFSet
|
|
|
457 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly
|
|
|
458
|
|
1300
|
459 data Omega-d : ( x : Ordinal ) → Set n where
|
|
|
460 iφ : Omega-d o∅
|
|
|
461 isuc : {x : Ordinal } → Omega-d x →
|
|
|
462 Omega-d (& ( Union (* x , (* x , * x ) ) ))
|
|
431
|
463
|
|
|
464 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
|
|
1300
|
465 -- We simply assumes Omega-d y has a maximum.
|
|
1091
|
466 --
|
|
431
|
467 -- This means that many of OD may not be HODs because of the & mapping divergence.
|
|
1300
|
468 -- We should have some axioms to prevent this .
|
|
1091
|
469 --
|
|
1300
|
470
|
|
|
471 Omega-od : OD
|
|
|
472 Omega-od = record { def = λ x → Omega-d x }
|
|
|
473
|
|
|
474 o∅<x : {x : Ordinal} → o∅ o≤ x
|
|
|
475 o∅<x {x} with trio< o∅ x
|
|
|
476 ... | tri< a ¬b ¬c = o<→≤ a
|
|
|
477 ... | tri≈ ¬a b ¬c = o≤-refl0 b
|
|
|
478 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
|
|
431
|
479
|
|
1300
|
480 ¬0=ux : {x : HOD} → ¬ o∅ ≡ & (Union ( x , ( x , x)))
|
|
|
481 ¬0=ux {x} eq = ⊥-elim ( o<¬≡ eq (ordtrans≤-< o∅<x (subst (λ k → k o< & (Union (x , (x , x)))) &iso (c<→o< lemma ) ))) where
|
|
|
482 lemma : Own (x , (x , x)) (& ( * (& x )))
|
|
|
483 lemma = record { owner = _ ; ao = case2 refl ; ox = subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (case1 refl) }
|
|
1297
|
484
|
|
1300
|
485 ux-2cases : {x y : HOD } → Union ( x , ( x , x)) ∋ y → ( x ≡ y ) ∨ ( x ∋ y )
|
|
|
486 ux-2cases {x} {y} record { owner = owner ; ao = (case1 eq) ; ox = ox } = case2 (subst (λ k → odef k (& y)) (trans (cong (*) eq) *iso) ox)
|
|
|
487 ux-2cases {x} {y} record { owner = owner ; ao = (case2 eq) ; ox = ox } with subst (λ k → odef k (& y)) (trans (cong (*) eq) *iso) ox
|
|
|
488 ... | case1 eq = case1 (sym (&≡&→≡ eq))
|
|
|
489 ... | case2 eq = case1 (sym (&≡&→≡ eq))
|
|
|
490
|
|
|
491 ux-transitve : {x y : HOD} → x ∋ y → Union ( x , ( x , x)) ∋ y
|
|
|
492 ux-transitve {x} {y} ox = record { owner = _ ; ao = case1 refl ; ox = subst (λ k → odef k (& y)) (sym *iso) ox }
|
|
|
493
|
|
|
494 --
|
|
|
495 -- Possible Ordinal Limit
|
|
|
496 --
|
|
|
497
|
|
|
498 -- our Ordinals is greater than Union ( x , ( x , x)) transitive closure
|
|
|
499 --
|
|
1297
|
500 record ODAxiom-ho< : Set (suc n) where
|
|
|
501 field
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|
|
502 omega : Ordinal
|
|
1300
|
503 ho< : {x : Ordinal } → Omega-d x → x o< omega
|
|
1297
|
504
|
|
|
505 postulate
|
|
|
506 odaxion-ho< : ODAxiom-ho<
|
|
|
507
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|
|
508 open ODAxiom-ho< odaxion-ho<
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|
|
509
|
|
1300
|
510 Omega : HOD
|
|
|
511 Omega = record { od = record { def = λ x → Omega-d x } ; odmax = omega ; <odmax = ho<}
|
|
431
|
512
|
|
1300
|
513 infinity∅ : Omega ∋ od∅
|
|
|
514 infinity∅ = subst (λ k → odef Omega k ) lemma iφ where
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515 lemma : o∅ ≡ & od∅
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516 lemma = let open ≡-Reasoning in begin
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517 o∅
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518 ≡⟨ sym &iso ⟩
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519 & ( * o∅ )
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520 ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
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521 & od∅
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522 ∎
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523
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524 infinity : (x : HOD) → Omega ∋ x → Omega ∋ Union (x , (x , x ))
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525 infinity x lt = subst (λ k → odef Omega k ) lemma (isuc {& x} lt) where
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526 lemma : & (Union (* (& x) , (* (& x) , * (& x))))
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527 ≡ & (Union (x , (x , x)))
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528 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
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431
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529
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1091
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530 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
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431
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531 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
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532 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
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533
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1091
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534 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
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431
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535 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
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536 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
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537
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1091
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538 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
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1096
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539 o<→c< lt {z} ox = ordtrans ox lt
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431
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540
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541 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
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1091
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542 ⊆→o< {x} {y} lt with trio< x y
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431
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543 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
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544 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
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1096
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545 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (o<-subst c (sym &iso) refl )
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431
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546 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
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547
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548 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
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549 ψiso {ψ} t refl = t
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550 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
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551 selection {ψ} {X} {y} = ⟪
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552 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ )
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553 , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ )
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554 ⟫
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555
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1091
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556 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
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431
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557 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
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558
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559 ---
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560 --- Power Set
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561 ---
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562 --- First consider ordinals in HOD
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563 ---
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564 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
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565 --
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566 --
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567 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
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568 ∩-≡ {a} {b} inc = record {
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569 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ;
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570 eq← = λ {x} x<a∩b → proj2 x<a∩b }
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571
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572 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
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1091
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573 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d
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574 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d
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431
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575
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576 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
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577 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
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1091
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578 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
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431
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579
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1284
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580 open import zf
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581
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1285
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582 record ODAxiom-sup : Set (suc n) where
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583 field
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584 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal -- required in Replace
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585 sup-o≤ : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal }
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586 → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤ sup-o A ψ
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587 sup-c≤ : (ψ : HOD → HOD) → {X x : HOD} → def (od X) (& x) → & (ψ x) o≤ (sup-o X (λ y X∋y → & (ψ (* y))))
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588 sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o≤ X lt )
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589
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590 -- sup-o may contradict
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591 -- If we have open monotonic function in Ordinal, there is no sup-o.
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592 -- for example, if we may have countable sequence of Ordinal, which contains some ordinal larger than any given Ordinal.
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593 -- This happens when we have a coutable model. In this case, we have to have codomain restriction in Replacement axiom.
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594 -- that is, Replacement axiom does not create new ZF set.
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595
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596 open ODAxiom-sup
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597
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598 ZFReplace : ODAxiom-sup → HOD → (HOD → HOD) → HOD
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599 ZFReplace os X ψ = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = rmax ; <odmax = rmax< } where
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600 rmax : Ordinal
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601 rmax = osuc ( sup-o os X (λ y X∋y → & (ψ (* y) )) )
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602 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< rmax
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603 rmax< {y} lt = subst (λ k → k o< rmax) r01 ( sup-o≤ os X (Replaced.az lt) ) where
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604 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
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605 r01 = sym (Replaced.x=ψz lt )
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606
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607 zf-replacement← : (os : ODAxiom-sup) → {ψ : HOD → HOD} (X x : HOD) → X ∋ x → ZFReplace os X ψ ∋ ψ x
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608 zf-replacement← os {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
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609 zf-replacement→ : (os : ODAxiom-sup ) → {ψ : HOD → HOD} (X x : HOD) → (lt : ZFReplace os X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
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610 zf-replacement→ os {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
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611
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1300
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612 isZF : (os : ODAxiom-sup) → IsZF HOD _∋_ _=h=_ od∅ _,_ Union Power Select (ZFReplace os) Omega
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1285
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613 isZF os = record {
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431
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614 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
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615 ; pair→ = pair→
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616 ; pair← = pair←
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617 ; union→ = union→
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618 ; union← = union←
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619 ; empty = empty
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1091
|
620 ; power→ = power→
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621 ; power← = power←
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622 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
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431
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623 ; ε-induction = ε-induction
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624 ; infinity∅ = infinity∅
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625 ; infinity = infinity
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626 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
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1285
|
627 ; replacement← = zf-replacement← os
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628 ; replacement→ = λ {ψ} → zf-replacement→ os {ψ}
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1091
|
629 }
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431
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630
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1285
|
631 HOD→ZF : ODAxiom-sup → ZF
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632 HOD→ZF os = record {
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1091
|
633 ZFSet = HOD
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634 ; _∋_ = _∋_
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635 ; _≈_ = _=h=_
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431
|
636 ; ∅ = od∅
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637 ; _,_ = _,_
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638 ; Union = Union
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639 ; Power = Power
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640 ; Select = Select
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1285
|
641 ; Replace = ZFReplace os
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|
1300
|
642 ; infinite = Omega
|
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1285
|
643 ; isZF = isZF os
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1091
|
644 }
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431
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645
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1091
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646
|
