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Comparison Functor all done.
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 02 Aug 2013 17:01:14 +0900
parents 3e05417b0876
children d6a6dd305da2
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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' : Monad 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 : Adjunction A B U^K F^K η^K ε^K )
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29 ( RK : MResolution A B T U^K F^K {η^K} {ε^K} {μ^K} Adj^K )
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30 where
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31
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32 open import adj-monad
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33
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34 T^K = U^K ○ F^K
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35
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36 μ^K' : NTrans A A (( U^K ○ F^K ) ○ ( U^K ○ F^K )) ( U^K ○ F^K )
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37 μ^K' = UεF A B U^K F^K ε^K
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38
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39 M : Monad A (U^K ○ F^K ) η^K μ^K'
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40 M = Adj2Monad A B {U^K} {F^K} {η^K} {ε^K} Adj^K
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41
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42 open import em-category {c₁} {c₂} {ℓ} {A} { U^K ○ F^K } { η^K } { μ^K' } { M }
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43
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44 open Functor
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45 open NTrans
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46 open Adjunction
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47 open MResolution
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48 open Eilenberg-Moore-Hom
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49
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50 emkobj : Obj B -> EMObj
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51 emkobj b = record {
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52 a = FObj U^K b ; phi = FMap U^K (TMap ε^K b) ; isAlgebra = record { identity = identity1 b; eval = eval1 b }
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53 } where
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54 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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55 identity1 b = let open ≈-Reasoning (A) in
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56 begin
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57 (FMap U^K (TMap ε^K b)) o TMap η^K (FObj U^K b)
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58 ≈⟨ IsAdjunction.adjoint1 (isAdjunction Adj^K) ⟩
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59 id1 A (FObj U^K b)
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60 ∎
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61
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62 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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63 eval1 b = let open ≈-Reasoning (A) in
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64 begin
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65 (FMap U^K (TMap ε^K b)) o TMap μ^K' (FObj U^K b)
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66 ≈⟨⟩
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67 (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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68 ≈⟨ sym (distr U^K) ⟩
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69 FMap U^K (B [ TMap ε^K b o (TMap ε^K ( FObj F^K (FObj U^K b))) ] )
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70 ≈⟨ fcong U^K (nat ε^K) ⟩ -- Horizontal composition
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71 FMap U^K (B [ TMap ε^K b o FMap F^K (FMap U^K (TMap ε^K b)) ] )
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72 ≈⟨ distr U^K ⟩
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73 (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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74 ≈⟨⟩
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75 (FMap U^K (TMap ε^K b)) o FMap T^K (FMap U^K (TMap ε^K b))
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76 ∎
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78
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79 open EMObj
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80 emkmap : {a b : Obj B} (f : Hom B a b) -> EMHom (emkobj a) (emkobj b)
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81 emkmap {a} {b} f = record { EMap = FMap U^K f ; homomorphism = homomorphism1 a b f
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82 } where
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83 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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84 homomorphism1 a b f = let open ≈-Reasoning (A) in
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85 begin
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86 (φ (emkobj b)) o FMap T^K (FMap U^K f)
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87 ≈⟨⟩
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88 FMap U^K (TMap ε^K b) o FMap U^K (FMap F^K (FMap U^K f))
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89 ≈⟨ sym (distr U^K) ⟩
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90 FMap U^K ( B [ TMap ε^K b o FMap F^K (FMap U^K f) ] )
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91 ≈⟨ sym (fcong U^K (nat ε^K)) ⟩
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92 FMap U^K ( B [ f o TMap ε^K a ] )
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93 ≈⟨ distr U^K ⟩
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94 (FMap U^K f) o FMap U^K (TMap ε^K a)
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95 ≈⟨⟩
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96 (FMap U^K f) o ( φ (emkobj a))
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97 ∎
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98
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99
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100 K^T : Functor B Eilenberg-MooreCategory
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101 K^T = record {
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102 FObj = emkobj
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103 ; FMap = emkmap
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104 ; isFunctor = record
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105 { ≈-cong = ≈-cong
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106 ; identity = identity
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107 ; distr = distr1
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108 }
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109 } where
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110 identity : {a : Obj B} → emkmap (id1 B a) ≗ EM-id {emkobj a}
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111 identity {a} = let open ≈-Reasoning (A) in
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112 begin
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113 EMap (emkmap (id1 B a))
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114 ≈⟨⟩
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115 FMap U^K (id1 B a)
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116 ≈⟨ IsFunctor.identity (isFunctor U^K) ⟩
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117 id1 A ( FObj U^K a )
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118 ≈⟨⟩
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119 EMap (EM-id {emkobj a})
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120 ∎
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parents: 122
diff changeset
121 ≈-cong : {a b : Obj B} -> {f g : Hom B a b} → B [ f ≈ g ] → emkmap f ≗ emkmap g
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
122 ≈-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
123 begin
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
124 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
125 ≈⟨ 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
126 EMap (emkmap g)
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
127 ∎
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
128 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
129 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
130 begin
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
131 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
132 ≈⟨ distr U^K ⟩
123
44c58c27d12d problems written Comparison Functor on EM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
133 EMap (emkmap g ∙ emkmap f)
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
134 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
135
128
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
136 Lemma-EM20 : { a b : Obj B} { f : Hom B a b } -> A [ FMap U^T ( FMap K^T f) ≈ FMap U^K f ]
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
137 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
138 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
139 FMap U^T ( FMap K^T f)
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
140 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
141 FMap U^K f
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
142 ∎
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
143
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
144 -- Lemma-EM21 : { a : Obj B} -> FObj U^T ( FObj K^T a) = FObj U^K a
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
145
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
146 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 ) ]
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
147 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
148 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
149 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
150 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
151 FMap U^K ( FMap F^K f)
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
152 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
153 EMap ( FMap F^T f )
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
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
156
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
157 -- Lemma-EM23 : { a b : Obj A} -> ( FObj K^T ( FObj F^K f) ) = ( FObj F^T f )
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
158
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
159 -- Lemma-EM24 : {a : Obj A } {b : Obj B} -> A [ TMap η^K (FObj U^K b) ≈ TMap η^K a ]
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
160 -- Lemma-EM24 = ?
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
161
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
162 Lemma-EM26 : {b : Obj B} -> A [ EMap (TMap ε^T ( FObj K^T b)) ≈ FMap U^K ( TMap ε^K b) ]
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
163 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
164 begin
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
165 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
166 ≈⟨⟩
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
167 FMap U^K ( TMap ε^K b)
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
276f33602700 Comparison Functor all done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
171