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1 open import Level
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2 open import Ordinals
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3 module BAlgbra {n : Level } (O : Ordinals {n}) where
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4
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5 open import zf
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6 open import logic
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7 import OrdUtil
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8 import OD
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9 import ODUtil
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10 import ODC
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11
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12 open import Relation.Nullary
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13 open import Relation.Binary
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14 open import Data.Empty
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15 open import Relation.Binary
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16 open import Relation.Binary.Core
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17 open import Relation.Binary.PropositionalEquality
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18 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ; _+_ to _n+_ )
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19
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20 open inOrdinal O
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21 open Ordinals.Ordinals O
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22 open Ordinals.IsOrdinals isOrdinal
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23 open Ordinals.IsNext isNext
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24 open OrdUtil O
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25 open ODUtil O
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26
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27 open OD O
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28 open OD.OD
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29 open ODAxiom odAxiom
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30 open HOD
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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 --_∩_ : ( A B : HOD ) → HOD
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37 --A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ;
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38 -- odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y)) }
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39
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450
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40 ∩-comm : { A B : HOD } → (A ∩ B) ≡ (B ∩ A)
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41 ∩-comm {A} {B} = ==→o≡ record { eq← = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ ; eq→ = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ }
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42
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43 _∪_ : ( A B : HOD ) → HOD
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44 A ∪ B = record { od = record { def = λ x → odef A x ∨ odef B x } ;
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45 odmax = omax (odmax A) (odmax B) ; <odmax = lemma } where
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46 lemma : {y : Ordinal} → odef A y ∨ odef B y → y o< omax (odmax A) (odmax B)
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47 lemma {y} (case1 a) = ordtrans (<odmax A a) (omax-x _ _)
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48 lemma {y} (case2 b) = ordtrans (<odmax B b) (omax-y _ _)
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49
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50 _＼_ : ( A B : HOD ) → HOD
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51 A ＼ B = record { od = record { def = λ x → odef A x ∧ ( ¬ ( odef B x ) ) }; odmax = odmax A ; <odmax = λ y → <odmax A (proj1 y) }
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52
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451
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53 ¬∅∋ : {x : HOD} → ¬ ( od∅ ∋ x )
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54 ¬∅∋ {x} = ¬x<0
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55
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56 [a-b]∩b=0 : { A B : HOD } → (A ＼ B) ∩ B ≡ od∅
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57 [a-b]∩b=0 {A} {B} = ==→o≡ record { eq← = λ lt → ⊥-elim ( ¬∅∋ (subst (λ k → odef od∅ k) (sym &iso) lt ))
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58 ; eq→ = λ {x} lt → ⊥-elim (proj2 (proj1 lt ) (proj2 lt)) }
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59
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431
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60 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
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61 ∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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62 lemma1 : {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x
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63 lemma1 {x} lt = lemma3 lt where
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64 lemma4 : {y : Ordinal} → odef (A , B) y ∧ odef (* y) x → ¬ (¬ ( odef A x ∨ odef B x) )
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65 lemma4 {y} z with proj1 z
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66 lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → odef k x ) *iso (proj2 z)) )
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67 lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → odef k x ) *iso (proj2 z)) )
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68 lemma3 : (((u : Ordinal ) → ¬ odef (A , B) u ∧ odef (* u) x) → ⊥) → odef (A ∪ B) x
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69 lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not) -- choice
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70 lemma2 : {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x
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71 lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) A
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72 ⟪ case1 refl , d→∋ A A∋x ⟫ )
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73 lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) &iso ( IsZF.union→ isZF (A , B) (* x) B
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74 ⟪ case2 refl , d→∋ B B∋x ⟫ )
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75
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76 ∩-Select : { A B : HOD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B )
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77 ∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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78 lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x
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79 lemma1 {x} lt = ⟪ proj1 lt , subst (λ k → odef B k ) &iso (proj2 (proj2 lt)) ⟫
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80 lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
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81 lemma2 {x} lt = ⟪ proj1 lt , ⟪ d→∋ A (proj1 lt) , d→∋ B (proj2 lt) ⟫ ⟫
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82
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83 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
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84 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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85 lemma1 : {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x
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86 lemma1 {x} lt with proj2 lt
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87 lemma1 {x} lt | case1 q∋x = case1 ⟪ proj1 lt , q∋x ⟫
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88 lemma1 {x} lt | case2 r∋x = case2 ⟪ proj1 lt , r∋x ⟫
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89 lemma2 : {x : Ordinal} → odef ((p ∩ q) ∪ (p ∩ r)) x → odef (p ∩ (q ∪ r)) x
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90 lemma2 {x} (case1 p∩q) = ⟪ proj1 p∩q , case1 (proj2 p∩q ) ⟫
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91 lemma2 {x} (case2 p∩r) = ⟪ proj1 p∩r , case2 (proj2 p∩r ) ⟫
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92
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93 dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r )
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94 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
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95 lemma1 : {x : Ordinal} → odef (p ∪ (q ∩ r)) x → odef ((p ∪ q) ∩ (p ∪ r)) x
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96 lemma1 {x} (case1 cp) = ⟪ case1 cp , case1 cp ⟫
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97 lemma1 {x} (case2 cqr) = ⟪ case2 (proj1 cqr) , case2 (proj2 cqr) ⟫
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98 lemma2 : {x : Ordinal} → odef ((p ∪ q) ∩ (p ∪ r)) x → odef (p ∪ (q ∩ r)) x
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99 lemma2 {x} lt with proj1 lt | proj2 lt
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100 lemma2 {x} lt | case1 cp | _ = case1 cp
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101 lemma2 {x} lt | _ | case1 cp = case1 cp
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102 lemma2 {x} lt | case2 cq | case2 cr = case2 ⟪ cq , cr ⟫
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103
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104 record IsBooleanAlgebra ( L : Set n)
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105 ( b1 : L )
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106 ( b0 : L )
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107 ( -_ : L → L )
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108 ( _+_ : L → L → L )
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109 ( _x_ : L → L → L ) : Set (suc n) where
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110 field
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111 +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c
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112 x-assoc : {a b c : L } → a x ( b x c ) ≡ (a x b) x c
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113 +-sym : {a b : L } → a + b ≡ b + a
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114 -sym : {a b : L } → a x b ≡ b x a
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115 +-aab : {a b : L } → a + ( a x b ) ≡ a
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116 x-aab : {a b : L } → a x ( a + b ) ≡ a
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117 +-dist : {a b c : L } → a + ( b x c ) ≡ ( a x b ) + ( a x c )
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118 x-dist : {a b c : L } → a x ( b + c ) ≡ ( a + b ) x ( a + c )
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119 a+0 : {a : L } → a + b0 ≡ a
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120 ax1 : {a : L } → a x b1 ≡ a
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121 a+-a1 : {a : L } → a + ( - a ) ≡ b1
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122 ax-a0 : {a : L } → a x ( - a ) ≡ b0
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123
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124 record BooleanAlgebra ( L : Set n) : Set (suc n) where
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125 field
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126 b1 : L
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127 b0 : L
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128 -_ : L → L
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129 _+_ : L → L → L
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130 _x_ : L → L → L
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131 isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _+_ _x_
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132
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