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1 module list-level where
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2
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3 open import Level
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4
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5
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6 postulate A : Set
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7 postulate B : Set
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8 postulate C : Set
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9
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10 postulate a : A
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11 postulate b : A
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12 postulate c : A
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13
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14
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15 infixr 40 _::_
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16 data List {a} (A : Set a) : Set a where
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17 [] : List A
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18 _::_ : A → List A → List A
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19
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20
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21 infixl 30 _++_
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22 _++_ : ∀ {a} {A : Set a} → List A → List A → List A
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23 [] ++ ys = ys
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24 (x :: xs) ++ ys = x :: (xs ++ ys)
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25
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26 l1 = a :: []
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27 l2 = a :: b :: a :: c :: []
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28
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29 l3 = l1 ++ l2
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30
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31 L1 = A :: []
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32 L2 = A :: B :: A :: C :: []
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33
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34 L3 = L1 ++ L2
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35
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36 data Node {a} ( A : Set a ) : Set a where
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37 leaf : A → Node A
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38 node : Node A → Node A → Node A
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39
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40 flatten : ∀{n} { A : Set n } → Node A → List A
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41 flatten ( leaf a ) = a :: []
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42 flatten ( node a b ) = flatten a ++ flatten b
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43
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44 n1 = node ( leaf a ) ( node ( leaf b ) ( leaf c ))
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45
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46 open import Relation.Binary.PropositionalEquality
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47
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48 infixr 20 _==_
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49
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50 data _==_ {n} {A : Set n} : List A → List A → Set n where
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51 reflection : {x : List A} → x == x
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52
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53 cong1 : ∀{a} {A : Set a } {b} { B : Set b } →
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54 ( f : List A → List B ) → {x : List A } → {y : List A} → x == y → f x == f y
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55 cong1 f reflection = reflection
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56
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57 eq-cons : ∀{n} {A : Set n} {x y : List A} ( a : A ) → x == y → ( a :: x ) == ( a :: y )
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58 eq-cons a z = cong1 ( λ x → ( a :: x) ) z
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59
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60 trans-list : ∀{n} {A : Set n} {x y z : List A} → x == y → y == z → x == z
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61 trans-list reflection reflection = reflection
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62
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63
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64 ==-to-≡ : ∀{n} {A : Set n} {x y : List A} → x == y → x ≡ y
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65 ==-to-≡ reflection = refl
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66
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67 list-id-l : { A : Set } → { x : List A} → [] ++ x == x
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68 list-id-l = reflection
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69
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70 list-id-r : { A : Set } → ( x : List A ) → x ++ [] == x
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71 list-id-r [] = reflection
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72 list-id-r (x :: xs) = eq-cons x ( list-id-r xs )
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73
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74 list-assoc : {A : Set } → ( xs ys zs : List A ) →
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75 ( ( xs ++ ys ) ++ zs ) == ( xs ++ ( ys ++ zs ) )
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76 list-assoc [] ys zs = reflection
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77 list-assoc (x :: xs) ys zs = eq-cons x ( list-assoc xs ys zs )
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78
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79
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80 module ==-Reasoning {n} (A : Set n ) where
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81
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82 infixr 2 _∎
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83 infixr 2 _==⟨_⟩_ _==⟨⟩_
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84 infix 1 begin_
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85
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86
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87 data _IsRelatedTo_ (x y : List A) :
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88 Set n where
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89 relTo : (x≈y : x == y ) → x IsRelatedTo y
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90
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91 begin_ : {x : List A } {y : List A} →
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92 x IsRelatedTo y → x == y
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93 begin relTo x≈y = x≈y
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94
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95 _==⟨_⟩_ : (x : List A ) {y z : List A} →
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96 x == y → y IsRelatedTo z → x IsRelatedTo z
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97 _ ==⟨ x≈y ⟩ relTo y≈z = relTo (trans-list x≈y y≈z)
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98
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99 _==⟨⟩_ : (x : List A ) {y : List A}
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100 → x IsRelatedTo y → x IsRelatedTo y
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101 _ ==⟨⟩ x≈y = x≈y
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102
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103 _∎ : (x : List A ) → x IsRelatedTo x
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104 _∎ _ = relTo reflection
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105
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106 lemma11 : ∀{n} (A : Set n) ( x : List A ) → x == x
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107 lemma11 A x = let open ==-Reasoning A in
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108 begin x ∎
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109
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110 ++-assoc : ∀{n} (L : Set n) ( xs ys zs : List L ) → (xs ++ ys) ++ zs == xs ++ (ys ++ zs)
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111 ++-assoc A [] ys zs = let open ==-Reasoning A in
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112 begin -- to prove ([] ++ ys) ++ zs == [] ++ (ys ++ zs)
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113 ( [] ++ ys ) ++ zs
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114 ==⟨ reflection ⟩
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115 ys ++ zs
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116 ==⟨ reflection ⟩
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117 [] ++ ( ys ++ zs )
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118 ∎
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119
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120 ++-assoc A (x :: xs) ys zs = let open ==-Reasoning A in
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121 begin -- to prove ((x :: xs) ++ ys) ++ zs == (x :: xs) ++ (ys ++ zs)
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122 ((x :: xs) ++ ys) ++ zs
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123 ==⟨ reflection ⟩
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124 (x :: (xs ++ ys)) ++ zs
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125 ==⟨ reflection ⟩
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126 x :: ((xs ++ ys) ++ zs)
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127 ==⟨ cong1 (_::_ x) (++-assoc A xs ys zs) ⟩
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128 x :: (xs ++ (ys ++ zs))
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129 ==⟨ reflection ⟩
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130 (x :: xs) ++ (ys ++ zs)
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131 ∎
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133
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134
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135 --data Bool : Set where
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136 -- true : Bool
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137 -- false : Bool
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138
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139
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140 --postulate Obj : Set
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142 --postulate Hom : Obj → Obj → Set
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143
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144
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145 --postulate id : { a : Obj } → Hom a a
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146
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147
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148 --infixr 80 _○_
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149 --postulate _○_ : { a b c : Obj } → Hom b c → Hom a b → Hom a c
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151 -- postulate axId1 : {a b : Obj} → ( f : Hom a b ) → f == id ○ f
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152 -- postulate axId2 : {a b : Obj} → ( f : Hom a b ) → f == f ○ id
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154 --assoc : { a b c d : Obj } →
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155 -- (f : Hom c d ) → (g : Hom b c) → (h : Hom a b) →
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156 -- (f ○ g) ○ h == f ○ ( g ○ h)
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157
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159 --a = Set
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161 -- ListObj : {A : Set} → List A
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162 -- ListObj = List Set
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164 -- ListHom : ListObj → ListObj → Set
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