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431
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1 {-# OPTIONS --allow-unsolved-metas #-}
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2
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3 open import Level
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4 open import Ordinals
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5 module OPair {n : Level } (O : Ordinals {n}) where
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6
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7 open import zf
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8 open import logic
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9 import OD
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10 import ODUtil
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11 import OrdUtil
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12
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13 open import Relation.Nullary
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14 open import Relation.Binary
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15 open import Data.Empty
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16 open import Relation.Binary
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17 open import Relation.Binary.Core
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18 open import Relation.Binary.PropositionalEquality
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19 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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20
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21 open OD O
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22 open OD.OD
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23 open OD.HOD
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24 open ODAxiom odAxiom
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25
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26 open Ordinals.Ordinals O
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27 open Ordinals.IsOrdinals isOrdinal
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28 open Ordinals.IsNext isNext
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29 open OrdUtil O
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30 open ODUtil O
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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 open _==_
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37
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38 <_,_> : (x y : HOD) → HOD
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39 < x , y > = (x , x ) , (x , y )
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40
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41 exg-pair : { x y : HOD } → (x , y ) =h= ( y , x )
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42 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
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43 left : {z : Ordinal} → odef (x , y) z → odef (y , x) z
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44 left (case1 t) = case2 t
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45 left (case2 t) = case1 t
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46 right : {z : Ordinal} → odef (y , x) z → odef (x , y) z
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47 right (case1 t) = case2 t
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48 right (case2 t) = case1 t
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49
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50 ord≡→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
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51 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong ( λ k → * k ) eq )
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52
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53 od≡→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
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54 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong ( λ k → & k ) eq )
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55
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56 eq-prod : { x x' y y' : HOD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
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57 eq-prod refl refl = refl
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58
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59 xx=zy→x=y : {x y z : HOD } → ( x , x ) =h= ( z , y ) → x ≡ y
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60 xx=zy→x=y {x} {y} eq with trio< (& x) (& y)
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61 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c with eq← eq {& y} (case2 refl)
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62 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
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63 xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
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64 xx=zy→x=y {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
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65 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c with eq← eq {& y} (case2 refl)
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66 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
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67 xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
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68
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69 prod-eq : { x x' y y' : HOD } → < x , y > =h= < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
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70 prod-eq {x} {x'} {y} {y'} eq = ⟪ lemmax , lemmay ⟫ where
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71 lemma2 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → z ≡ y
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72 lemma2 {x} {y} {z} eq = trans (sym (xx=zy→x=y lemma3 )) ( xx=zy→x=y eq ) where
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73 lemma3 : ( x , x ) =h= ( y , z )
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74 lemma3 = ==-trans eq exg-pair
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75 lemma1 : {x y : HOD } → ( x , x ) =h= ( y , y ) → x ≡ y
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76 lemma1 {x} {y} eq with eq← eq {& y} (case2 refl)
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77 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
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78 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
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79 lemma4 : {x y z : HOD } → ( x , y ) =h= ( x , z ) → y ≡ z
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80 lemma4 {x} {y} {z} eq with eq← eq {& z} (case2 refl)
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81 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
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82 ... | refl with lemma2 (==-sym eq )
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83 ... | refl = refl
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84 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
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85 lemmax : x ≡ x'
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86 lemmax with eq→ eq {& (x , x)} (case1 refl)
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87 lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x')
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88 lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
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89 ... | refl = lemma1 (ord→== s )
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90 lemmay : y ≡ y'
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91 lemmay with lemmax
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92 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
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93 ... | eq1 = lemma4 (ord→== (cong (λ k → & k ) eq1 ))
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94
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95 --
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96 -- unlike ordered pair, ZFProduct is not a HOD
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97
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98 data ord-pair : (p : Ordinal) → Set n where
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99 pair : (x y : Ordinal ) → ord-pair ( & ( < * x , * y > ) )
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100
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101 ZFProduct : OD
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102 ZFProduct = record { def = λ x → ord-pair x }
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103
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104 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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105 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
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106 -- eq-pair refl refl = HE.refl
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107
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108 pi1 : { p : Ordinal } → ord-pair p → Ordinal
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109 pi1 ( pair x y) = x
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110
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111 π1 : { p : HOD } → def ZFProduct (& p) → HOD
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112 π1 lt = * (pi1 lt )
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113
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114 pi2 : { p : Ordinal } → ord-pair p → Ordinal
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115 pi2 ( pair x y ) = y
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116
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117 π2 : { p : HOD } → def ZFProduct (& p) → HOD
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118 π2 lt = * (pi2 lt )
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119
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120 op-cons : { ox oy : Ordinal } → def ZFProduct (& ( < * ox , * oy > ))
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121 op-cons {ox} {oy} = pair ox oy
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122
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123 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
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124 def-subst df refl refl = df
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125
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126 p-cons : ( x y : HOD ) → def ZFProduct (& ( < x , y >))
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127 p-cons x y = def-subst {_} {_} {ZFProduct} {& (< x , y >)} (pair (& x) ( & y )) refl (
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128 let open ≡-Reasoning in begin
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129 & < * (& x) , * (& y) >
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130 ≡⟨ cong₂ (λ j k → & < j , k >) *iso *iso ⟩
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131 & < x , y >
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132 ∎ )
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133
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134 op-iso : { op : Ordinal } → (q : ord-pair op ) → & < * (pi1 q) , * (pi2 q) > ≡ op
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135 op-iso (pair ox oy) = refl
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136
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137 p-iso : { x : HOD } → (p : def ZFProduct (& x) ) → < π1 p , π2 p > ≡ x
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138 p-iso {x} p = ord≡→≡ (op-iso p)
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139
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140 p-pi1 : { x y : HOD } → (p : def ZFProduct (& < x , y >) ) → π1 p ≡ x
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141 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
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142
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143 p-pi2 : { x y : HOD } → (p : def ZFProduct (& < x , y >) ) → π2 p ≡ y
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144 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
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145
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146 ω-pair : {x y : HOD} → {m : Ordinal} → & x o< next m → & y o< next m → & (x , y) o< next m
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147 ω-pair lx ly = next< (omax<nx lx ly ) ho<
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148
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149 ω-opair : {x y : HOD} → {m : Ordinal} → & x o< next m → & y o< next m → & < x , y > o< next m
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150 ω-opair {x} {y} {m} lx ly = lemma0 where
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151 lemma0 : & < x , y > o< next m
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152 lemma0 = osucprev (begin
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153 osuc (& < x , y >)
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154 <⟨ osuc<nx ho< ⟩
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155 next (omax (& (x , x)) (& (x , y)))
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156 ≡⟨ cong (λ k → next k) (sym ( omax≤ _ _ pair-xx<xy )) ⟩
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157 next (osuc (& (x , y)))
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158 ≡⟨ sym (nexto≡) ⟩
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159 next (& (x , y))
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160 ≤⟨ x<ny→≤next (ω-pair lx ly) ⟩
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161 next m
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162 ∎ ) where
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163 open o≤-Reasoning O
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164
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165 _⊗_ : (A B : HOD) → HOD
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166 A ⊗ B = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) ))
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167
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168 product→ : {A B a b : HOD} → A ∋ a → B ∋ b → ( A ⊗ B ) ∋ < a , b >
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169 product→ {A} {B} {a} {b} A∋a B∋b = λ t → t (& (Replace A (λ a → < a , b >)))
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170 ⟪ lemma1 , subst (λ k → odef k (& < a , b >)) (sym *iso) lemma2 ⟫ where
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171 lemma1 : odef (Replace B (λ b₁ → Replace A (λ a₁ → < a₁ , b₁ >))) (& (Replace A (λ a₁ → < a₁ , b >)))
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172 lemma1 = replacement← B b B∋b
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173 lemma2 : odef (Replace A (λ a₁ → < a₁ , b >)) (& < a , b >)
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174 lemma2 = replacement← A a A∋a
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175
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176 x<nextA : {A x : HOD} → A ∋ x → & x o< next (odmax A)
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177 x<nextA {A} {x} A∋x = ordtrans (c<→o< {x} {A} A∋x) ho<
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178
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179 A<Bnext : {A B x : HOD} → & A o< & B → A ∋ x → & x o< next (odmax B)
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180 A<Bnext {A} {B} {x} lt A∋x = osucprev (begin
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181 osuc (& x)
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182 <⟨ osucc (c<→o< A∋x) ⟩
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183 osuc (& A)
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184 <⟨ osucc lt ⟩
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185 osuc (& B)
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186 <⟨ osuc<nx ho< ⟩
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187 next (odmax B)
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188 ∎ ) where open o≤-Reasoning O
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189
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190 ZFP : (A B : HOD) → HOD
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191 ZFP A B = record { od = record { def = λ x → ord-pair x ∧ ((p : ord-pair x ) → odef A (pi1 p) ∧ odef B (pi2 p) )} ;
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192 odmax = omax (next (odmax A)) (next (odmax B)) ; <odmax = λ {y} px → lemma y px }
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193 where
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194 lemma : (y : Ordinal) → ( ord-pair y ∧ ((p : ord-pair y) → odef A (pi1 p) ∧ odef B (pi2 p))) → y o< omax (next (odmax A)) (next (odmax B))
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195 lemma y lt with proj1 lt
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196 lemma p lt | pair x y with trio< (& A) (& B)
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197 lemma p lt | pair x y | tri< a ¬b ¬c = ordtrans (ω-opair (A<Bnext a (subst (λ k → odef A k ) (sym &iso)
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198 (proj1 (proj2 lt (pair x y))))) (lemma1 (proj2 (proj2 lt (pair x y))))) (omax-y _ _ ) where
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199 lemma1 : odef B y → & (* y) o< next (HOD.odmax B)
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200 lemma1 lt = x<nextA {B} (d→∋ B lt)
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201 lemma p lt | pair x y | tri≈ ¬a b ¬c = ordtrans (ω-opair (x<nextA {A} (d→∋ A ((proj1 (proj2 lt (pair x y)))))) lemma2 ) (omax-x _ _ ) where
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202 lemma2 : & (* y) o< next (HOD.odmax A)
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203 lemma2 = ordtrans ( subst (λ k → & (* y) o< k ) (sym b) (c<→o< (d→∋ B ((proj2 (proj2 lt (pair x y))))))) ho<
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204 lemma p lt | pair x y | tri> ¬a ¬b c = ordtrans (ω-opair (x<nextA {A} (d→∋ A ((proj1 (proj2 lt (pair x y))))))
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205 (A<Bnext c (subst (λ k → odef B k ) (sym &iso) (proj2 (proj2 lt (pair x y)))))) (omax-x _ _ )
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206
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207 ZFP⊆⊗ : {A B : HOD} {x : Ordinal} → odef (ZFP A B) x → odef (A ⊗ B) x
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208 ZFP⊆⊗ {A} {B} {px} ⟪ (pair x y) , p2 ⟫ = product→ (d→∋ A (proj1 (p2 (pair x y) ))) (d→∋ B (proj2 (p2 (pair x y) )))
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209
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210 -- axiom of choice required
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211 -- ⊗⊆ZFP : {A B x : HOD} → ( A ⊗ B ) ∋ x → def ZFProduct (& x)
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212 -- ⊗⊆ZFP {A} {B} {x} lt = subst (λ k → ord-pair (& k )) {!!} op-cons
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213
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