Mercurial > hg > Members > kono > Proof > category
annotate nat.agda @ 73:b75b5792bd81
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 26 Jul 2013 15:52:24 +0900 |
| parents | cbc30519e961 |
| children | 49e6eb3ef9c0 |
| rev | line source |
|---|---|
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parents:
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1 |
| 0 | 2 |
| 3 -- Monad | |
| 4 -- Category A | |
| 5 -- A = Category | |
| 22 | 6 -- Functor T : A → A |
| 0 | 7 --T(a) = t(a) |
| 8 --T(f) = tf(f) | |
| 9 | |
| 2 | 10 open import Category -- https://github.com/konn/category-agda |
| 0 | 11 open import Level |
| 56 | 12 --open import Category.HomReasoning |
| 13 open import HomReasoning | |
| 14 open import cat-utility | |
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15 open import Category.Cat |
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16 |
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17 module nat { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } |
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18 { T : Functor A A } |
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19 { η : NTrans A A identityFunctor T } |
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20 { μ : NTrans A A (T ○ T) T } |
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21 { M : Monad A T η μ } |
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22 { K : Kleisli A T η μ M } where |
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23 |
| 0 | 24 |
| 1 | 25 --T(g f) = T(g) T(f) |
| 26 | |
| 31 | 27 open Functor |
| 22 | 28 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 } |
| 29 → A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ] | |
| 30 Lemma1 = \t → IsFunctor.distr ( isFunctor t ) | |
| 0 | 31 |
| 32 | |
| 7 | 33 open NTrans |
| 1 | 34 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A} |
| 22 | 35 → (μ : NTrans A A F G) → {a b : Obj A} { f : Hom A a b } |
| 30 | 36 → A [ A [ FMap G f o TMap μ a ] ≈ A [ TMap μ b o FMap F f ] ] |
| 22 | 37 Lemma2 = \n → IsNTrans.naturality ( isNTrans n ) |
| 0 | 38 |
| 22 | 39 -- η : 1_A → T |
| 40 -- μ : TT → T | |
| 0 | 41 -- μ(a)η(T(a)) = a |
| 42 -- μ(a)T(η(a)) = a | |
| 43 -- μ(a)(μ(T(a))) = μ(a)T(μ(a)) | |
| 44 | |
| 45 | |
| 2 | 46 open Monad |
| 47 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 48 { T : Functor A A } | |
| 7 | 49 { η : NTrans A A identityFunctor T } |
| 50 { μ : NTrans A A (T ○ T) T } | |
| 22 | 51 { a : Obj A } → |
| 2 | 52 ( M : Monad A T η μ ) |
| 30 | 53 → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] |
| 22 | 54 Lemma3 = \m → IsMonad.assoc ( isMonad m ) |
| 2 | 55 |
| 56 | |
| 57 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b} | |
| 22 | 58 → A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ] |
| 59 Lemma4 = \a → IsCategory.identityL ( Category.isCategory a ) | |
| 0 | 60 |
| 3 | 61 Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} |
| 62 { T : Functor A A } | |
| 7 | 63 { η : NTrans A A identityFunctor T } |
| 64 { μ : NTrans A A (T ○ T) T } | |
| 22 | 65 { a : Obj A } → |
| 3 | 66 ( M : Monad A T η μ ) |
| 30 | 67 → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 68 Lemma5 = \m → IsMonad.unity1 ( isMonad m ) |
| 3 | 69 |
| 70 Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 71 { T : Functor A A } | |
| 7 | 72 { η : NTrans A A identityFunctor T } |
| 73 { μ : NTrans A A (T ○ T) T } | |
| 22 | 74 { a : Obj A } → |
| 3 | 75 ( M : Monad A T η μ ) |
| 30 | 76 → A [ A [ TMap μ a o (FMap T (TMap η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 77 Lemma6 = \m → IsMonad.unity2 ( isMonad m ) |
| 3 | 78 |
| 79 -- T = M x A | |
| 0 | 80 -- nat of η |
| 81 -- g ○ f = μ(c) T(g) f | |
| 82 -- η(b) ○ f = f | |
| 83 -- f ○ η(a) = f | |
| 22 | 84 -- h ○ (g ○ f) = (h ○ g) ○ f |
| 0 | 85 |
| 7 | 86 |
| 22 | 87 lemma12 : {c₁ c₂ ℓ : Level} (L : Category c₁ c₂ ℓ) { a b c : Obj L } → |
| 88 ( x : Hom L c a ) → ( y : Hom L b c ) → L [ L [ x o y ] ≈ L [ x o y ] ] | |
| 18 | 89 lemma12 L x y = |
| 90 let open ≈-Reasoning ( L ) in | |
| 91 begin L [ x o y ] ∎ | |
| 11 | 92 |
| 4 | 93 open Kleisli |
| 22 | 94 -- η(b) ○ f = f |
| 73 | 95 Lemma7 : { a : Obj A } { b : Obj A } ( f : Hom A a ( FObj T b) ) |
| 96 → A [ join K (TMap η b) f ≈ f ] | |
| 97 Lemma7 {_} {b} f = | |
| 98 let open ≈-Reasoning (A) in | |
| 21 | 99 begin |
| 73 | 100 join K (TMap η b) f |
| 21 | 101 ≈⟨ refl-hom ⟩ |
| 73 | 102 A [ TMap μ b o A [ FMap T ((TMap η b)) o f ] ] |
| 103 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ | |
| 104 A [ A [ TMap μ b o FMap T ((TMap η b)) ] o f ] | |
| 105 ≈⟨ car ( IsMonad.unity2 ( isMonad ( monad K )) ) ⟩ | |
| 106 A [ id (FObj T b) o f ] | |
| 107 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ | |
| 21 | 108 f |
| 109 ∎ | |
| 7 | 110 |
| 22 | 111 -- f ○ η(a) = f |
| 73 | 112 Lemma8 : { a : Obj A } { b : Obj A } |
| 22 | 113 ( f : Hom A a ( FObj T b) ) |
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114 → A [ join K f (TMap η a) ≈ f ] |
| 73 | 115 Lemma8 {a} {b} f = |
| 22 | 116 begin |
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117 join K f (TMap η a) |
| 22 | 118 ≈⟨ refl-hom ⟩ |
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119 A [ TMap μ b o A [ FMap T f o (TMap η a) ] ] |
| 66 | 120 ≈⟨ cdr ( nat η ) ⟩ |
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121 A [ TMap μ b o A [ (TMap η ( FObj T b)) o f ] ] |
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122 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ |
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parents:
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123 A [ A [ TMap μ b o (TMap η ( FObj T b)) ] o f ] |
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124 ≈⟨ car ( IsMonad.unity1 ( isMonad ( monad K )) ) ⟩ |
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125 A [ id (FObj T b) o f ] |
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126 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ |
| 22 | 127 f |
| 128 ∎ where | |
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129 open ≈-Reasoning (A) |
| 5 | 130 |
| 22 | 131 -- h ○ (g ○ f) = (h ○ g) ○ f |
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132 Lemma9 : { a b c d : Obj A } |
| 73 | 133 ( h : Hom A c ( FObj T d) ) |
| 23 | 134 ( g : Hom A b ( FObj T c) ) |
| 73 | 135 ( f : Hom A a ( FObj T b) ) |
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136 → A [ join K h (join K g f) ≈ join K ( join K h g) f ] |
| 73 | 137 Lemma9 {a} {b} {c} {d} h g f = |
| 24 | 138 begin |
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139 join K h (join K g f) |
| 30 | 140 ≈⟨⟩ |
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141 join K h ( ( TMap μ c o ( FMap T g o f ) ) ) |
| 28 | 142 ≈⟨ refl-hom ⟩ |
| 30 | 143 ( TMap μ d o ( FMap T h o ( TMap μ c o ( FMap T g o f ) ) ) ) |
| 28 | 144 ≈⟨ cdr ( cdr ( assoc )) ⟩ |
| 30 | 145 ( TMap μ d o ( FMap T h o ( ( TMap μ c o FMap T g ) o f ) ) ) |
| 28 | 146 ≈⟨ assoc ⟩ --- ( f o ( g o h ) ) = ( ( f o g ) o h ) |
| 30 | 147 ( ( TMap μ d o FMap T h ) o ( (TMap μ c o FMap T g ) o f ) ) |
| 25 | 148 ≈⟨ assoc ⟩ |
| 30 | 149 ( ( ( TMap μ d o FMap T h ) o (TMap μ c o FMap T g ) ) o f ) |
| 28 | 150 ≈⟨ car (sym assoc) ⟩ |
| 30 | 151 ( ( TMap μ d o ( FMap T h o ( TMap μ c o FMap T g ) ) ) o f ) |
| 28 | 152 ≈⟨ car ( cdr (assoc) ) ⟩ |
| 30 | 153 ( ( TMap μ d o ( ( FMap T h o TMap μ c ) o FMap T g ) ) o f ) |
| 28 | 154 ≈⟨ car assoc ⟩ |
| 30 | 155 ( ( ( TMap μ d o ( FMap T h o TMap μ c ) ) o FMap T g ) o f ) |
| 28 | 156 ≈⟨ car (car ( cdr ( begin |
| 30 | 157 ( FMap T h o TMap μ c ) |
| 66 | 158 ≈⟨ nat μ ⟩ |
| 30 | 159 ( TMap μ (FObj T d) o FMap T (FMap T h) ) |
| 25 | 160 ∎ |
| 161 ))) ⟩ | |
| 30 | 162 ( ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) ) o FMap T g ) o f ) |
| 28 | 163 ≈⟨ car (sym assoc) ⟩ |
| 30 | 164 ( ( TMap μ d o ( ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) o FMap T g ) ) o f ) |
| 28 | 165 ≈⟨ car ( cdr (sym assoc) ) ⟩ |
| 30 | 166 ( ( TMap μ d o ( TMap μ ( FObj T d) o ( FMap T ( FMap T h ) o FMap T g ) ) ) o f ) |
| 28 | 167 ≈⟨ car ( cdr (cdr (sym (distr T )))) ⟩ |
| 30 | 168 ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 28 | 169 ≈⟨ car assoc ⟩ |
| 30 | 170 ( ( ( TMap μ d o TMap μ ( FObj T d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 171 ≈⟨ car ( car ( |
| 27 | 172 begin |
| 30 | 173 ( TMap μ d o TMap μ (FObj T d) ) |
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174 ≈⟨ IsMonad.assoc ( isMonad M) ⟩ |
| 30 | 175 ( TMap μ d o FMap T (TMap μ d) ) |
| 27 | 176 ∎ |
| 177 )) ⟩ | |
| 30 | 178 ( ( ( TMap μ d o FMap T ( TMap μ d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 179 ≈⟨ car (sym assoc) ⟩ |
| 30 | 180 ( ( TMap μ d o ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 24 | 181 ≈⟨ sym assoc ⟩ |
| 30 | 182 ( TMap μ d o ( ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) o f ) ) |
| 28 | 183 ≈⟨ cdr ( car ( sym ( distr T ))) ⟩ |
| 30 | 184 ( TMap μ d o ( FMap T ( ( ( TMap μ d ) o ( FMap T h o g ) ) ) o f ) ) |
| 23 | 185 ≈⟨ refl-hom ⟩ |
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186 join K ( ( TMap μ d o ( FMap T h o g ) ) ) f |
| 23 | 187 ≈⟨ refl-hom ⟩ |
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188 join K ( join K h g) f |
| 24 | 189 ∎ where open ≈-Reasoning (A) |
| 3 | 190 |
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191 record KHom (a : Obj A) (b : Obj A) |
| 70 | 192 : Set (suc (c₂ ⊔ c₁)) where |
| 193 field | |
| 194 KMap : Hom A a ( FObj T b ) | |
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195 |
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196 K-id : {a : Obj A} → KHom a a |
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197 K-id {a = a} = record { KMap = TMap η a } |
| 56 | 198 |
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199 open import Relation.Binary.Core |
| 73 | 200 open KHom |
| 56 | 201 |
| 73 | 202 _⋍_ : { a : Obj A } { b : Obj A } (f g : KHom a b ) -> Set ℓ -- (suc ((c₂ ⊔ c₁) ⊔ ℓ)) |
| 203 _⋍_ {a} {b} f g = A [ KMap f ≈ KMap g ] | |
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204 |
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205 _*_ : { a b : Obj A } → { c : Obj A } → |
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206 ( KHom b c) → ( KHom a b) → KHom a c |
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207 _*_ {a} {b} {c} g f = record { KMap = join K {a} {b} {c} (KMap g) (KMap f) } |
| 70 | 208 |
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209 isKleisliCategory : IsCategory ( Obj A ) KHom _⋍_ _*_ K-id |
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210 isKleisliCategory = record { isEquivalence = isEquivalence |
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211 ; identityL = KidL |
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212 ; identityR = KidR |
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213 ; o-resp-≈ = Ko-resp |
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214 ; associative = Kassoc |
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215 } |
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216 where |
| 73 | 217 open ≈-Reasoning (A) |
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218 isEquivalence : { a b : Obj A } -> |
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219 IsEquivalence {_} {_} {KHom a b} _⋍_ |
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220 isEquivalence {C} {D} = |
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221 record { refl = λ {F} → ⋍-refl {F} |
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222 ; sym = λ {F} {G} → ⋍-sym {F} {G} |
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223 ; trans = λ {F} {G} {H} → ⋍-trans {F} {G} {H} |
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224 } where |
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225 ⋍-refl : {F : KHom C D} → F ⋍ F |
| 73 | 226 ⋍-refl = refl-hom |
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stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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227 ⋍-sym : {F G : KHom C D} → F ⋍ G → G ⋍ F |
| 73 | 228 ⋍-sym = sym |
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72
cbc30519e961
stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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229 ⋍-trans : {F G H : KHom C D} → F ⋍ G → G ⋍ H → F ⋍ H |
| 73 | 230 ⋍-trans = trans-hom |
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72
cbc30519e961
stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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231 KidL : {C D : Obj A} -> {f : KHom C D} → (K-id * f) ⋍ f |
| 73 | 232 KidL {_} {_} {f} = Lemma7 (KMap f) |
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72
cbc30519e961
stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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233 KidR : {C D : Obj A} -> {f : KHom C D} → (f * K-id) ⋍ f |
| 73 | 234 KidR {_} {_} {f} = Lemma8 (KMap f) |
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72
cbc30519e961
stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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235 Ko-resp : {a b c : Obj A} -> {f g : KHom a b } → {h i : KHom b c } → |
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71
709c1bde54dc
Kleisli category problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
70
diff
changeset
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236 f ⋍ g → h ⋍ i → (h * f) ⋍ (i * g) |
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69
84a150c980ce
generalized distr and assco1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
68
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changeset
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237 Ko-resp = {!!} |
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72
cbc30519e961
stack overflow solved by moving implicit parameters to module parameters
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
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238 Kassoc : {a b c d : Obj A} -> {f : KHom c d } → {g : KHom b c } → {h : KHom a b } → |
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71
709c1bde54dc
Kleisli category problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
70
diff
changeset
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239 (f * (g * h)) ⋍ ((f * g) * h) |
| 73 | 240 Kassoc {_} {_} {_} {_} {f} {g} {h} = Lemma9 (KMap f) (KMap g) (KMap h) |
| 3 | 241 |
| 56 | 242 |