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annotate comparison-em.agda @ 787:ca5eba647990

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 18 Apr 2019 20:07:22 +0900
parents a5f2ca67e7c5
children
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1 -- -- -- -- -- -- -- --
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2 -- Comparison Functor of Eilenberg-Moore Category
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3 -- defines U^K and F^K as a resolution of Monad
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4 -- checks Adjointness
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5 --
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6 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 -- -- -- -- -- -- -- --
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8
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9 open import Category -- https://github.com/konn/category-agda
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10 open import Level
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11 --open import Category.HomReasoning
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12 open import HomReasoning
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13 open import cat-utility
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14 open import Category.Cat
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15 open import Relation.Binary.Core
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16
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17 module comparison-em
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18 { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }
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19 { T : Functor A A }
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20 { η : NTrans A A identityFunctor T }
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21 { μ : NTrans A A (T ○ T) T }
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22 { M' : IsMonad A T η μ }
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23 {c₁' c₂' ℓ' : Level} ( B : Category c₁' c₂' ℓ' )
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24 { U^K : Functor B A } { F^K : Functor A B }
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25 { η^K : NTrans A A identityFunctor ( U^K ○ F^K ) }
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26 { ε^K : NTrans B B ( F^K ○ U^K ) identityFunctor }
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27 { μ^K : NTrans A A (( U^K ○ F^K ) ○ ( U^K ○ F^K )) ( U^K ○ F^K ) }
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28 ( Adj^K : IsAdjunction A B U^K F^K η^K ε^K )
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29 where
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30
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31 open import adj-monad
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32
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33 T^K = U^K ○ F^K
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34
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35 μ^K' : NTrans A A (( U^K ○ F^K ) ○ ( U^K ○ F^K )) ( U^K ○ F^K )
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36 μ^K' = UεF A B U^K F^K ε^K
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37
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38 M : IsMonad A (U^K ○ F^K ) η^K μ^K'
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39 M = Monad.isMonad ( Adj2Monad A B ( record { U = U^K; F = F^K ; η = η^K ; ε = ε^K ; isAdjunction = Adj^K } ) )
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40
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41 open import em-category {c₁} {c₂} {ℓ} {A} { U^K ○ F^K } { η^K } { μ^K' } { M }
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42
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43 open Functor
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44 open NTrans
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45 open Adjunction
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46 open MResolution
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47 open EMHom
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48
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49 emkobj : Obj B → EMObj
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50 emkobj b = record {
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51 obj = FObj U^K b ; φ = FMap U^K (TMap ε^K b) ; isAlgebra = record { identity = identity1 b; eval = eval1 b }
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52 } where
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53 identity1 : (b : Obj B) → A [ A [ (FMap U^K (TMap ε^K b)) o TMap η^K (FObj U^K b) ] ≈ id1 A (FObj U^K b) ]
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54 identity1 b = let open ≈-Reasoning (A) in
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55 begin
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56 (FMap U^K (TMap ε^K b)) o TMap η^K (FObj U^K b)
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57 ≈⟨ IsAdjunction.adjoint1 Adj^K ⟩
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58 id1 A (FObj U^K b)
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59 ∎
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60
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61 eval1 : (b : Obj B) → A [ A [ (FMap U^K (TMap ε^K b)) o TMap μ^K' (FObj U^K b) ] ≈ A [ (FMap U^K (TMap ε^K b)) o FMap T^K (FMap U^K (TMap ε^K b)) ] ]
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62 eval1 b = let open ≈-Reasoning (A) in
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63 begin
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64 (FMap U^K (TMap ε^K b)) o TMap μ^K' (FObj U^K b)
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65 ≈⟨⟩
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66 (FMap U^K (TMap ε^K b)) o FMap U^K (TMap ε^K ( FObj F^K (FObj U^K b)))
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67 ≈⟨ sym (distr U^K) ⟩
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68 FMap U^K (B [ TMap ε^K b o (TMap ε^K ( FObj F^K (FObj U^K b))) ] )
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69 ≈⟨ fcong U^K (nat ε^K) ⟩ -- Horizontal composition
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70 FMap U^K (B [ TMap ε^K b o FMap F^K (FMap U^K (TMap ε^K b)) ] )
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71 ≈⟨ distr U^K ⟩
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72 (FMap U^K (TMap ε^K b)) o FMap U^K (FMap F^K (FMap U^K (TMap ε^K b)))
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73 ≈⟨⟩
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74 (FMap U^K (TMap ε^K b)) o FMap T^K (FMap U^K (TMap ε^K b))
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75 ∎
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76
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77
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78 open EMObj
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79 emkmap : {a b : Obj B} (f : Hom B a b) → EMHom (emkobj a) (emkobj b)
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80 emkmap {a} {b} f = record { EMap = FMap U^K f ; homomorphism = homomorphism1 a b f
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81 } where
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82 homomorphism1 : (a b : Obj B) (f : Hom B a b) → A [ A [ (φ (emkobj b)) o FMap T^K (FMap U^K f) ] ≈ A [ (FMap U^K f) o (φ (emkobj a)) ] ]
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83 homomorphism1 a b f = let open ≈-Reasoning (A) in
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84 begin
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85 (φ (emkobj b)) o FMap T^K (FMap U^K f)
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86 ≈⟨⟩
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87 FMap U^K (TMap ε^K b) o FMap U^K (FMap F^K (FMap U^K f))
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88 ≈⟨ sym (distr U^K) ⟩
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89 FMap U^K ( B [ TMap ε^K b o FMap F^K (FMap U^K f) ] )
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90 ≈⟨ sym (fcong U^K (nat ε^K)) ⟩
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91 FMap U^K ( B [ f o TMap ε^K a ] )
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92 ≈⟨ distr U^K ⟩
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93 (FMap U^K f) o FMap U^K (TMap ε^K a)
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94 ≈⟨⟩
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95 (FMap U^K f) o ( φ (emkobj a))
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96 ∎
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97
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98
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99 K^T : Functor B Eilenberg-MooreCategory
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100 K^T = record {
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101 FObj = emkobj
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102 ; FMap = emkmap
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103 ; isFunctor = record
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104 { ≈-cong = ≈-cong
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105 ; identity = identity
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106 ; distr = distr1
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107 }
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108 } where
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109 identity : {a : Obj B} → emkmap (id1 B a) ≗ EM-id {emkobj a}
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110 identity {a} = let open ≈-Reasoning (A) in
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111 begin
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112 EMap (emkmap (id1 B a))
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113 ≈⟨⟩
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114 FMap U^K (id1 B a)
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115 ≈⟨ IsFunctor.identity (isFunctor U^K) ⟩
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116 id1 A ( FObj U^K a )
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117 ≈⟨⟩
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118 EMap (EM-id {emkobj a})
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119 ∎
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parents: 128
diff changeset
120 ≈-cong : {a b : Obj B} → {f g : Hom B a b} → B [ f ≈ g ] → emkmap f ≗ emkmap g
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
121 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning (A) in
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
122 begin
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
123 EMap (emkmap f)
127
3e05417b0876 Comparison Functor for Eilenberg-Moore Category is constructed.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
124 ≈⟨ IsFunctor.≈-cong (isFunctor U^K) f≈g ⟩
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
125 EMap (emkmap g)
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
126 ∎
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
127 distr1 : {a b c : Obj B} {f : Hom B a b} {g : Hom B b c} → ( (emkmap (B [ g o f ])) ≗ (emkmap g ∙ emkmap f) )
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
128 distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning (A) in
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
129 begin
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
130 EMap (emkmap (B [ g o f ] ))
127
3e05417b0876 Comparison Functor for Eilenberg-Moore Category is constructed.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
131 ≈⟨ distr U^K ⟩
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
132 EMap (emkmap g ∙ emkmap f)
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
133 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
134
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
135 Lemma-EM20 : { a b : Obj B} { f : Hom B a b } → A [ FMap U^T ( FMap K^T f) ≈ FMap U^K f ]
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
136 Lemma-EM20 {a} {b} {f} = let open ≈-Reasoning (A) in
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
137 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
138 FMap U^T ( FMap K^T f)
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
139 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
140 FMap U^K f
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
141 ∎
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
142
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
143 -- Lemma-EM21 : { a : Obj B} → FObj U^T ( FObj K^T a) = FObj U^K a
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
144
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
145 Lemma-EM22 : { a b : Obj A} { f : Hom A a b } → A [ EMap ( FMap K^T ( FMap F^K f) ) ≈ EMap ( FMap F^T f ) ]
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
146 Lemma-EM22 {a} {b} {f} = let open ≈-Reasoning (A) in
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
147 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
148 EMap ( FMap K^T ( FMap F^K f) )
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
149 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
150 FMap U^K ( FMap F^K f)
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
151 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
152 EMap ( FMap F^T f )
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
153 ∎
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
154
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
155
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
156 -- Lemma-EM23 : { a b : Obj A} → ( FObj K^T ( FObj F^K f) ) = ( FObj F^T f )
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
157
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
158 -- Lemma-EM24 : {a : Obj A } {b : Obj B} → A [ TMap η^K (FObj U^K b) ≈ TMap η^K a ]
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
159 -- Lemma-EM24 = ?
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
160
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
161 Lemma-EM26 : {b : Obj B} → A [ EMap (TMap ε^T ( FObj K^T b)) ≈ FMap U^K ( TMap ε^K b) ]
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
162 Lemma-EM26 {b} = let open ≈-Reasoning (A) in
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
163 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
164 EMap (TMap ε^T ( FObj K^T b))
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
165 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
166 FMap U^K ( TMap ε^K b)
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
167 ∎
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
168
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
169
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
170