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1 module nat where
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2
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3 -- Monad
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4 -- Category A
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5 -- A = Category
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6 -- Functor T : A -> A
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7 --T(a) = t(a)
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8 --T(f) = tf(f)
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9
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10 open import Category -- https://github.com/konn/category-agda
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11 open import Level
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12 open Functor
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13
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14 --T(g f) = T(g) T(f)
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15
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16 Lemma1 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} (T : Functor A A) -> {a b c : Obj A} {g : Hom A b c} { f : Hom A a b }
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17 -> A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ]
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18 Lemma1 = \t -> IsFunctor.distr ( isFunctor t )
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19
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20 -- F(f)
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21 -- F(a) ----> F(b)
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22 -- | |
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23 -- |t(a) |t(b) G(f)t(a) = t(b)F(f)
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24 -- | |
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25 -- v v
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26 -- G(a) ----> G(b)
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27 -- G(f)
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28
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29 record IsNTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (D : Category c₁ c₂ ℓ) (C : Category c₁′ c₂′ ℓ′)
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30 ( F G : Functor D C )
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31 (Trans : (A : Obj D) → Hom C (FObj F A) (FObj G A))
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32 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where
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33 field
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34 naturality : {a b : Obj D} {f : Hom D a b}
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35 → C [ C [ ( FMap G f ) o ( Trans a ) ] ≈ C [ (Trans b ) o (FMap F f) ] ]
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36 -- uniqness : {d : Obj D}
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37 -- → C [ Trans d ≈ Trans d ]
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38
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39
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40 record NTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (domain : Category c₁ c₂ ℓ) (codomain : Category c₁′ c₂′ ℓ′) (F G : Functor domain codomain )
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41 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where
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42 field
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43 Trans : (A : Obj domain) → Hom codomain (FObj F A) (FObj G A)
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44 isNTrans : IsNTrans domain codomain F G Trans
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45
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46 open NTrans
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47 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A}
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48 -> (μ : NTrans A A F G) -> {a b : Obj A} { f : Hom A a b }
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49 -> A [ A [ FMap G f o Trans μ a ] ≈ A [ Trans μ b o FMap F f ] ]
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50 Lemma2 = \n -> IsNTrans.naturality ( isNTrans n )
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51
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52 open import Category.Cat
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53
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54 -- η : 1_A -> T
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55 -- μ : TT -> T
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56 -- μ(a)η(T(a)) = a
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57 -- μ(a)T(η(a)) = a
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58 -- μ(a)(μ(T(a))) = μ(a)T(μ(a))
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59
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60 record IsMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
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61 ( T : Functor A A )
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62 ( η : NTrans A A identityFunctor T )
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63 ( μ : NTrans A A (T ○ T) T)
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64 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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65 field
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66 assoc : {a : Obj A} -> A [ A [ Trans μ a o Trans μ ( FObj T a ) ] ≈ A [ Trans μ a o FMap T (Trans μ a) ] ]
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67 unity1 : {a : Obj A} -> A [ A [ Trans μ a o Trans η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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68 unity2 : {a : Obj A} -> A [ A [ Trans μ a o (FMap T (Trans η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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69
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70 record Monad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (T : Functor A A) (η : NTrans A A identityFunctor T) (μ : NTrans A A (T ○ T) T)
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71 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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72 eta : NTrans A A identityFunctor T
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73 eta = η
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74 mu : NTrans A A (T ○ T) T
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75 mu = μ
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76 field
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77 isMonad : IsMonad A T η μ
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78
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79 open Monad
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80 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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81 { T : Functor A A }
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82 { η : NTrans A A identityFunctor T }
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83 { μ : NTrans A A (T ○ T) T }
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84 { a : Obj A } ->
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85 ( M : Monad A T η μ )
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86 -> A [ A [ Trans μ a o Trans μ ( FObj T a ) ] ≈ A [ Trans μ a o FMap T (Trans μ a) ] ]
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87 Lemma3 = \m -> IsMonad.assoc ( isMonad m )
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88
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89
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90 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b}
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91 -> A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ]
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92 Lemma4 = \a -> IsCategory.identityL ( Category.isCategory a )
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93
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94 Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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95 { T : Functor A A }
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96 { η : NTrans A A identityFunctor T }
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97 { μ : NTrans A A (T ○ T) T }
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98 { a : Obj A } ->
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99 ( M : Monad A T η μ )
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100 -> A [ A [ Trans μ a o Trans η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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101 Lemma5 = \m -> IsMonad.unity1 ( isMonad m )
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102
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103 Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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104 { T : Functor A A }
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105 { η : NTrans A A identityFunctor T }
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106 { μ : NTrans A A (T ○ T) T }
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107 { a : Obj A } ->
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108 ( M : Monad A T η μ )
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109 -> A [ A [ Trans μ a o (FMap T (Trans η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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110 Lemma6 = \m -> IsMonad.unity2 ( isMonad m )
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111
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112 -- T = M x A
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113 -- nat of η
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114 -- g ○ f = μ(c) T(g) f
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115 -- h ○ (g ○ f) = (h ○ g) ○ f
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116 -- η(b) ○ f = f
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117 -- f ○ η(a) = f
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118
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119 record Kleisli { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ )
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120 ( T : Functor A A )
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121 ( η : NTrans A A identityFunctor T )
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122 ( μ : NTrans A A (T ○ T) T )
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123 ( M : Monad A T η μ ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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124 monad : Monad A T η μ
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125 monad = M
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126 join : { a b : Obj A } -> ( c : Obj A ) ->
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127 ( Hom A b ( FObj T c )) -> ( Hom A a ( FObj T b)) -> Hom A a ( FObj T c )
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128 join c g f = A [ Trans μ c o A [ FMap T g o f ] ]
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129
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130 open import Relation.Binary renaming (Trans to Trans1 )
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131 open import Relation.Binary.Core renaming (Trans to Trans1 )
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132 open import Relation.Binary.PropositionalEquality using (_≡_; refl)
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133
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134 module ≈-Reasoning where
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135
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136 -- The code in Relation.Binary.EqReasoning cannot handle
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137 -- heterogeneous equalities, hence the code duplication here.
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138
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139 refl-hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A }
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140 { x y z : Hom A a b } →
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141 A [ x ≈ x ]
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142 refl-hom = \a -> IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory a ))
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143
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144 trans-hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A }
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145 { x y z : Hom A a b } →
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146 A [ x ≈ y ] → A [ y ≈ z ] → A [ x ≈ z ]
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147 trans-hom a b c = ( IsEquivalence.trans (IsCategory.isEquivalence ( Category.isCategory a ))) b c
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148
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149 infixr 2 _∎_
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150 infixr 2 _≈⟨_!_⟩_
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151 infix 1 begin_
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152
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153 data IsRelatedTo {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } ( x y : Hom A a b ) :
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154 Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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155 relTo : (x≈y : A [ x ≈ y ] ) → IsRelatedTo A x y
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156
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157 begin_ : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} {a b : Obj A } { x y : Hom A a b} →
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158 IsRelatedTo A x y → A [ x ≈ y ]
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159 begin relTo x≈y = x≈y
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160
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161 _≈⟨_!_⟩_ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) -> {a b : Obj A }
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162 ( x : Hom A a b ) → { y z : Hom A a b } →
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163 A [ x ≈ y ] → IsRelatedTo A y z → IsRelatedTo A x z
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164 a ≈⟨ _ ! x≈y ⟩ relTo y≈z = relTo ( trans-hom a x≈y y≈z )
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165
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166 _∎_ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } (x : Hom A a b) → IsRelatedTo A x x
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167 a ∎ _ = relTo ( refl-hom a )
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168
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169 open Kleisli
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170 Lemma7 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ->
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171 { T : Functor A A }
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172 { η : NTrans A A identityFunctor T }
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173 { μ : NTrans A A (T ○ T) T }
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174 { a b : Obj A }
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175 { f : Hom A a ( FObj T b) }
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176 { M : Monad A T η μ }
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177 ( K : Kleisli A T η μ M)
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178 -> A [ join K b (Trans η b) f ≈ f ]
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179 Lemma7 c k = {!!}
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180
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181
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182
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183
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184 Lemma8 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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185 { T : Functor A A }
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186 { η : NTrans A A identityFunctor T }
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187 { μ : NTrans A A (T ○ T) T }
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188 { a b : Obj A }
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189 { f : Hom A a ( FObj T b) }
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190 { M : Monad A T η μ }
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191 ( K : Kleisli A T η μ M)
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192 -> A [ join K b f (Trans η a) ≈ f ]
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193 Lemma8 k = {!!}
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194
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195 Lemma9 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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196 { T : Functor A A }
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197 { η : NTrans A A identityFunctor T }
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198 { μ : NTrans A A (T ○ T) T }
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199 { a b c d : Obj A }
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200 { f : Hom A a ( FObj T b) }
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201 { g : Hom A b ( FObj T c) }
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202 { h : Hom A c ( FObj T d) }
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203 { M : Monad A T η μ }
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204 ( K : Kleisli A T η μ M)
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205 -> A [ join K d h (join K c g f) ≈ join K d ( join K d h g) f ]
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206 Lemma9 k = {!!}
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207
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208
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209
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210
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211
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212 -- Kleisli :
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213 -- Kleisli = record { Hom =
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214 -- ; Hom = _⟶_
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215 -- ; Id = IdProd
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216 -- ; _o_ = _∘_
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217 -- ; _≈_ = _≈_
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218 -- ; isCategory = record { isEquivalence = record { refl = λ {φ} → ≈-refl {φ = φ}
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219 -- ; sym = λ {φ ψ} → ≈-symm {φ = φ} {ψ}
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220 -- ; trans = λ {φ ψ σ} → ≈-trans {φ = φ} {ψ} {σ}
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221 -- }
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222 -- ; identityL = identityL
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223 -- ; identityR = identityR
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224 -- ; o-resp-≈ = o-resp-≈
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225 -- ; associative = associative
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226 -- }
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227 -- }
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