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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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1218
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5 module ZProduct {n : Level } (O : Ordinals {n}) where
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431
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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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1098
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95 prod-≡ : { x x' y y' : HOD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
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96 prod-≡ eq = prod-eq (ord→== (cong (&) eq ))
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97
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431
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98 --
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1098
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99 -- unlike ordered pair, ZFPair is not a HOD
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431
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100
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101 data ord-pair : (p : Ordinal) → Set n where
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102 pair : (x y : Ordinal ) → ord-pair ( & ( < * x , * y > ) )
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103
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1098
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104 ZFPair : OD
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105 ZFPair = record { def = λ x → ord-pair x }
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431
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106
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107 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
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108 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
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109 -- eq-pair refl refl = HE.refl
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110
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111 pi1 : { p : Ordinal } → ord-pair p → Ordinal
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112 pi1 ( pair x y) = x
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113
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1098
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114 π1 : { p : HOD } → def ZFPair (& p) → HOD
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431
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115 π1 lt = * (pi1 lt )
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116
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117 pi2 : { p : Ordinal } → ord-pair p → Ordinal
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118 pi2 ( pair x y ) = y
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119
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1098
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120 π2 : { p : HOD } → def ZFPair (& p) → HOD
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431
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121 π2 lt = * (pi2 lt )
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122
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1098
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123 op-cons : ( ox oy : Ordinal ) → def ZFPair (& ( < * ox , * oy > ))
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124 op-cons ox oy = pair ox oy
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431
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125
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126 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
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127 def-subst df refl refl = df
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128
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1098
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129 p-cons : ( x y : HOD ) → def ZFPair (& ( < x , y >))
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130 p-cons x y = def-subst {_} {_} {ZFPair} {& (< x , y >)} (pair (& x) ( & y )) refl (
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431
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131 let open ≡-Reasoning in begin
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132 & < * (& x) , * (& y) >
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133 ≡⟨ cong₂ (λ j k → & < j , k >) *iso *iso ⟩
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134 & < x , y >
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135 ∎ )
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136
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137 op-iso : { op : Ordinal } → (q : ord-pair op ) → & < * (pi1 q) , * (pi2 q) > ≡ op
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138 op-iso (pair ox oy) = refl
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139
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1098
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140 p-iso : { x : HOD } → (p : def ZFPair (& x) ) → < π1 p , π2 p > ≡ x
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431
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141 p-iso {x} p = ord≡→≡ (op-iso p)
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142
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1098
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143 p-pi1 : { x y : HOD } → (p : def ZFPair (& < x , y >) ) → π1 p ≡ x
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431
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144 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
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145
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1098
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146 p-pi2 : { x y : HOD } → (p : def ZFPair (& < x , y >) ) → π2 p ≡ y
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431
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147 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
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148
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149 _⊗_ : (A B : HOD) → HOD
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150 A ⊗ B = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) ))
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151
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152 product→ : {A B a b : HOD} → A ∋ a → B ∋ b → ( A ⊗ B ) ∋ < a , b >
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1096
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153 product→ {A} {B} {a} {b} A∋a B∋b = record { owner = _ ; ao = lemma1 ; ox = subst (λ k → odef k _) (sym *iso) lemma2 } where
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431
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154 lemma1 : odef (Replace B (λ b₁ → Replace A (λ a₁ → < a₁ , b₁ >))) (& (Replace A (λ a₁ → < a₁ , b >)))
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155 lemma1 = replacement← B b B∋b
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156 lemma2 : odef (Replace A (λ a₁ → < a₁ , b >)) (& < a , b >)
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157 lemma2 = replacement← A a A∋a
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158
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1098
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159 data ZFProduct (A B : HOD) : (p : Ordinal) → Set n where
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160 ab-pair : {a b : Ordinal } → odef A a → odef B b → ZFProduct A B ( & ( < * a , * b > ) )
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161
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431
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162 ZFP : (A B : HOD) → HOD
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1098
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163 ZFP A B = record { od = record { def = λ x → ZFProduct A B x }
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1169
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164 ; odmax = odmax ( A ⊗ B ) ; <odmax = λ {y} px → <odmax ( A ⊗ B ) (lemma0 px) }
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431
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165 where
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1169
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166 lemma0 : {A B : HOD} {x : Ordinal} → ZFProduct A B x → odef (A ⊗ B) x
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167 lemma0 {A} {B} {px} ( ab-pair {a} {b} ax by ) = product→ (d→∋ A ax) (d→∋ B by)
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1098
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168
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169 ZFP→ : {A B a b : HOD} → A ∋ a → B ∋ b → ZFP A B ∋ < a , b >
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170 ZFP→ {A} {B} {a} {b} aa bb = subst (λ k → ZFProduct A B k ) (cong₂ (λ j k → & < j , k >) *iso *iso ) ( ab-pair aa bb )
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431
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171
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1104
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172 zπ1 : {A B : HOD} → {x : Ordinal } → odef (ZFP A B) x → Ordinal
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173 zπ1 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb) = a
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174
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175 zp1 : {A B : HOD} → {x : Ordinal } → (zx : odef (ZFP A B) x) → odef A (zπ1 zx)
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176 zp1 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb ) = aa
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177
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178 zπ2 : {A B : HOD} → {x : Ordinal } → odef (ZFP A B) x → Ordinal
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179 zπ2 (ab-pair {a} {b} aa bb) = b
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180
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181 zp2 : {A B : HOD} → {x : Ordinal } → (zx : odef (ZFP A B) x) → odef B (zπ2 zx)
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182 zp2 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb ) = bb
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183
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184 zp-iso : { A B : HOD } → {x : Ordinal } → (p : odef (ZFP A B) x ) → & < * (zπ1 p) , * (zπ2 p) > ≡ x
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185 zp-iso {A} {B} {_} (ab-pair {a} {b} aa bb) = refl
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186
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1216
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187 zp-iso1 : { A B : HOD } → {a b : Ordinal } → (p : odef (ZFP A B) (& < * a , * b > )) → (* (zπ1 p) ≡ (* a)) ∧ (* (zπ2 p) ≡ (* b))
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188 zp-iso1 {A} {B} {a} {b} pab = prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) zz11) ) where
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189 zz11 : & < * (zπ1 pab) , * (zπ2 pab) > ≡ & < * a , * b >
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190 zz11 = zp-iso pab
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191
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1223
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192 zp-iso0 : { A B : HOD } → {a b : Ordinal } → (p : odef (ZFP A B) (& < * a , * b > )) → (zπ1 p ≡ a) ∧ (zπ2 p ≡ b)
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193 zp-iso0 {A} {B} {a} {b} pab = ⟪ subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj1 (zp-iso1 pab) ))
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194 , subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj2 (zp-iso1 pab) ) ) ⟫
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195
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431
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196 ZFP⊆⊗ : {A B : HOD} {x : Ordinal} → odef (ZFP A B) x → odef (A ⊗ B) x
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1098
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197 ZFP⊆⊗ {A} {B} {px} ( ab-pair {a} {b} ax by ) = product→ (d→∋ A ax) (d→∋ B by)
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198
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199 ⊗⊆ZFPair : {A B x : HOD} → ( A ⊗ B ) ∋ x → def ZFPair (& x)
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200 ⊗⊆ZFPair {A} {B} {x} record { owner = owner ; ao = record { z = a ; az = aa ; x=ψz = x=ψa } ; ox = ox } = zfp01 where
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201 zfp02 : Replace A (λ z → < z , * a >) ≡ * owner
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202 zfp02 = subst₂ ( λ j k → j ≡ k ) *iso refl (sym (cong (*) x=ψa ))
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203 zfp01 : def ZFPair (& x)
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204 zfp01 with subst (λ k → odef k (& x) ) (sym zfp02) ox
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205 ... | record { z = b ; az = ab ; x=ψz = x=ψb } = subst (λ k → def ZFPair k) (cong (&) zfp00) (op-cons b a ) where
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206 zfp00 : < * b , * a > ≡ x
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207 zfp00 = sym ( subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) x=ψb) )
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431
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208
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1098
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209 ⊗⊆ZFP : {A B x : HOD} → ( A ⊗ B ) ∋ x → odef (ZFP A B) (& x)
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210 ⊗⊆ZFP {A} {B} {x} record { owner = owner ; ao = record { z = a ; az = ba ; x=ψz = x=ψa } ; ox = ox } = zfp01 where
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211 zfp02 : Replace A (λ z → < z , * a >) ≡ * owner
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212 zfp02 = subst₂ ( λ j k → j ≡ k ) *iso refl (sym (cong (*) x=ψa ))
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213 zfp01 : odef (ZFP A B) (& x)
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214 zfp01 with subst (λ k → odef k (& x) ) (sym zfp02) ox
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215 ... | record { z = b ; az = ab ; x=ψz = x=ψb } = subst (λ k → ZFProduct A B k ) (sym x=ψb) (ab-pair ab ba)
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431
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216
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1105
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217 ZFPproj1 : {A B X : HOD} → X ⊆ ZFP A B → HOD
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218 ZFPproj1 {A} {B} {X} X⊆P = Replace' X ( λ x px → * (zπ1 (X⊆P px) ))
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219
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220 ZFPproj2 : {A B X : HOD} → X ⊆ ZFP A B → HOD
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221 ZFPproj2 {A} {B} {X} X⊆P = Replace' X ( λ x px → * (zπ2 (X⊆P px) ))
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1098
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222
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1218
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223 ZFProj1-iso : {P Q : HOD} {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ1 p ≡ a
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224 ZFProj1-iso {P} {Q} {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
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225 ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) a=c)
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1105
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226
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1218
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227 ZFProj2-iso : {P Q : HOD} {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ2 p ≡ b
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228 ZFProj2-iso {P} {Q} {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
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229 ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) b=d)
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1105
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230
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1219
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231 ZFP∩ : {A B C : HOD} → ( ZFP (A ∩ B) C ≡ ZFP A C ∩ ZFP B C ) ∧ ( ZFP C (A ∩ B) ≡ ZFP C A ∩ ZFP C B )
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232 proj1 (ZFP∩ {A} {B} {C} ) = ==→o≡ record { eq→ = zfp00 ; eq← = zfp01 } where
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233 zfp00 : {x : Ordinal} → ZFProduct (A ∩ B) C x → odef (ZFP A C ∩ ZFP B C) x
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234 zfp00 (ab-pair ⟪ pa , pb ⟫ qx) = ⟪ ab-pair pa qx , ab-pair pb qx ⟫
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235 zfp01 : {x : Ordinal} → odef (ZFP A C ∩ ZFP B C) x → ZFProduct (A ∩ B) C x
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1220
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236 zfp01 {x} ⟪ p , q ⟫ = subst (λ k → ZFProduct (A ∩ B) C k) zfp07 ( ab-pair (zfp02 ⟪ p , q ⟫ ) (zfp04 q) ) where
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237 zfp05 : & < * (zπ1 p) , * (zπ2 p) > ≡ x
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238 zfp05 = zp-iso p
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239 zfp06 : & < * (zπ1 q) , * (zπ2 q) > ≡ x
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240 zfp06 = zp-iso q
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241 zfp07 : & < * (zπ1 p) , * (zπ2 q) > ≡ x
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242 zfp07 = trans (cong (λ k → & < k , * (zπ2 q) > )
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243 (proj1 (prod-≡ (subst₂ _≡_ *iso *iso (cong (*) (trans zfp05 (sym (zfp06)))))))) zfp06
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1219
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244 zfp02 : {x : Ordinal } → (acx : odef (ZFP A C ∩ ZFP B C) x) → odef (A ∩ B) (zπ1 (proj1 acx))
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1220
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245 zfp02 {.(& < * _ , * _ >)} ⟪ ab-pair {a} {b} ax bx , bcx ⟫ = ⟪ ax , zfp03 bcx refl ⟫ where
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1219
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246 zfp03 : {x : Ordinal } → (bc : odef (ZFP B C) x) → x ≡ (& < * a , * b >) → odef B (zπ1 (ab-pair {A} {C} ax bx))
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1220
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247 zfp03 (ab-pair {a1} {b1} x x₁) eq = subst (λ k → odef B k ) zfp08 x where
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248 zfp08 : a1 ≡ a
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249 zfp08 = subst₂ _≡_ &iso &iso (cong (&) (proj1 (prod-≡ (subst₂ _≡_ *iso *iso (cong (*) eq)))))
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1219
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250 zfp04 : {x : Ordinal } (acx : odef (ZFP B C) x )→ odef C (zπ2 acx)
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1220
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251 zfp04 (ab-pair x x₁) = x₁
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252 proj2 (ZFP∩ {A} {B} {C} ) = ==→o≡ record { eq→ = zfp00 ; eq← = zfp01 } where
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253 zfp00 : {x : Ordinal} → ZFProduct C (A ∩ B) x → odef (ZFP C A ∩ ZFP C B) x
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254 zfp00 (ab-pair qx ⟪ pa , pb ⟫ ) = ⟪ ab-pair qx pa , ab-pair qx pb ⟫
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255 zfp01 : {x : Ordinal} → odef (ZFP C A ∩ ZFP C B ) x → ZFProduct C (A ∩ B) x
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256 zfp01 {x} ⟪ p , q ⟫ = subst (λ k → ZFProduct C (A ∩ B) k) zfp07 ( ab-pair (zfp04 p) (zfp02 ⟪ p , q ⟫ ) ) where
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257 zfp05 : & < * (zπ1 p) , * (zπ2 p) > ≡ x
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258 zfp05 = zp-iso p
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259 zfp06 : & < * (zπ1 q) , * (zπ2 q) > ≡ x
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260 zfp06 = zp-iso q
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261 zfp07 : & < * (zπ1 p) , * (zπ2 q) > ≡ x
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262 zfp07 = trans (cong (λ k → & < * (zπ1 p) , k > )
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263 (sym (proj2 (prod-≡ (subst₂ _≡_ *iso *iso (cong (*) (trans zfp05 (sym (zfp06))))))))) zfp05
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264 zfp02 : {x : Ordinal } → (acx : odef (ZFP C A ∩ ZFP C B ) x) → odef (A ∩ B) (zπ2 (proj2 acx))
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265 zfp02 {.(& < * _ , * _ >)} ⟪ bcx , ab-pair {b} {a} ax bx ⟫ = ⟪ zfp03 bcx refl , bx ⟫ where
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266 zfp03 : {x : Ordinal } → (bc : odef (ZFP C A ) x) → x ≡ (& < * b , * a >) → odef A (zπ2 (ab-pair {C} {B} ax bx ))
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267 zfp03 (ab-pair {b1} {a1} x x₁) eq = subst (λ k → odef A k ) zfp08 x₁ where
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268 zfp08 : a1 ≡ a
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269 zfp08 = subst₂ _≡_ &iso &iso (cong (&) (proj2 (prod-≡ (subst₂ _≡_ *iso *iso (cong (*) eq)))))
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270 zfp04 : {x : Ordinal } (acx : odef (ZFP C A ) x )→ odef C (zπ1 acx)
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271 zfp04 (ab-pair x x₁) = x
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1219
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272
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1224
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273 open import BAlgebra O
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274
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275 ZFP\Q : {P Q p : HOD} → (( ZFP P Q ＼ ZFP p Q ) ≡ ZFP (P ＼ p) Q ) ∧ (( ZFP P Q ＼ ZFP P p ) ≡ ZFP P (Q ＼ p) )
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276 ZFP\Q {P} {Q} {p} = ⟪ ==→o≡ record { eq→ = ty70 ; eq← = ty71 } , ==→o≡ record { eq→ = ty73 ; eq← = ty75 } ⟫ where
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277 ty70 : {x : Ordinal } → odef ( ZFP P Q ＼ ZFP p Q ) x → odef (ZFP (P ＼ p) Q) x
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278 ty70 ⟪ ab-pair {a} {b} Pa pb , npq ⟫ = ab-pair ty72 pb where
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279 ty72 : odef (P ＼ p ) a
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280 ty72 = ⟪ Pa , (λ pa → npq (ab-pair pa pb ) ) ⟫
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281 ty71 : {x : Ordinal } → odef (ZFP (P ＼ p) Q) x → odef ( ZFP P Q ＼ ZFP p Q ) x
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282 ty71 (ab-pair {a} {b} ⟪ Pa , npa ⟫ Qb) = ⟪ ab-pair Pa Qb
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283 , (λ pab → npa (subst (λ k → odef p k) (proj1 (zp-iso0 pab)) (zp1 pab)) ) ⟫
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284 ty73 : {x : Ordinal } → odef ( ZFP P Q ＼ ZFP P p ) x → odef (ZFP P (Q ＼ p) ) x
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285 ty73 ⟪ ab-pair {a} {b} pa Qb , npq ⟫ = ab-pair pa ty72 where
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286 ty72 : odef (Q ＼ p ) b
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287 ty72 = ⟪ Qb , (λ qb → npq (ab-pair pa qb ) ) ⟫
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288 ty75 : {x : Ordinal } → odef (ZFP P (Q ＼ p) ) x → odef ( ZFP P Q ＼ ZFP P p ) x
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289 ty75 (ab-pair {a} {b} Pa ⟪ Qb , nqb ⟫ ) = ⟪ ab-pair Pa Qb
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290 , (λ pab → nqb (subst (λ k → odef p k) (proj2 (zp-iso0 pab)) (zp2 pab)) ) ⟫
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1219
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291
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292
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293
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294
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295
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