Mercurial > hg > Members > kono > Proof > category
annotate nat.agda @ 56:83ff8d48fdca
add unitility
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 24 Jul 2013 20:47:28 +0900 |
| parents | 17b8bafebad7 |
| children | c6f66c21230c |
| rev | line source |
|---|---|
| 0 | 1 module nat where |
| 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 | |
| 0 | 15 |
| 1 | 16 --T(g f) = T(g) T(f) |
| 17 | |
| 31 | 18 open Functor |
| 22 | 19 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 } |
| 20 → A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ] | |
| 21 Lemma1 = \t → IsFunctor.distr ( isFunctor t ) | |
| 0 | 22 |
| 23 | |
| 7 | 24 open NTrans |
| 1 | 25 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A} |
| 22 | 26 → (μ : NTrans A A F G) → {a b : Obj A} { f : Hom A a b } |
| 30 | 27 → A [ A [ FMap G f o TMap μ a ] ≈ A [ TMap μ b o FMap F f ] ] |
| 22 | 28 Lemma2 = \n → IsNTrans.naturality ( isNTrans n ) |
| 0 | 29 |
| 30 open import Category.Cat | |
| 31 | |
| 22 | 32 -- η : 1_A → T |
| 33 -- μ : TT → T | |
| 0 | 34 -- μ(a)η(T(a)) = a |
| 35 -- μ(a)T(η(a)) = a | |
| 36 -- μ(a)(μ(T(a))) = μ(a)T(μ(a)) | |
| 37 | |
| 38 | |
| 2 | 39 open Monad |
| 40 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 41 { T : Functor A A } | |
| 7 | 42 { η : NTrans A A identityFunctor T } |
| 43 { μ : NTrans A A (T ○ T) T } | |
| 22 | 44 { a : Obj A } → |
| 2 | 45 ( M : Monad A T η μ ) |
| 30 | 46 → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] |
| 22 | 47 Lemma3 = \m → IsMonad.assoc ( isMonad m ) |
| 2 | 48 |
| 49 | |
| 50 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b} | |
| 22 | 51 → A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ] |
| 52 Lemma4 = \a → IsCategory.identityL ( Category.isCategory a ) | |
| 0 | 53 |
| 3 | 54 Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} |
| 55 { T : Functor A A } | |
| 7 | 56 { η : NTrans A A identityFunctor T } |
| 57 { μ : NTrans A A (T ○ T) T } | |
| 22 | 58 { a : Obj A } → |
| 3 | 59 ( M : Monad A T η μ ) |
| 30 | 60 → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 61 Lemma5 = \m → IsMonad.unity1 ( isMonad m ) |
| 3 | 62 |
| 63 Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 64 { T : Functor A A } | |
| 7 | 65 { η : NTrans A A identityFunctor T } |
| 66 { μ : NTrans A A (T ○ T) T } | |
| 22 | 67 { a : Obj A } → |
| 3 | 68 ( M : Monad A T η μ ) |
| 30 | 69 → A [ A [ TMap μ a o (FMap T (TMap η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 70 Lemma6 = \m → IsMonad.unity2 ( isMonad m ) |
| 3 | 71 |
| 72 -- T = M x A | |
| 0 | 73 -- nat of η |
| 74 -- g ○ f = μ(c) T(g) f | |
| 75 -- η(b) ○ f = f | |
| 76 -- f ○ η(a) = f | |
| 22 | 77 -- h ○ (g ○ f) = (h ○ g) ○ f |
| 0 | 78 |
| 7 | 79 |
| 22 | 80 lemma12 : {c₁ c₂ ℓ : Level} (L : Category c₁ c₂ ℓ) { a b c : Obj L } → |
| 81 ( x : Hom L c a ) → ( y : Hom L b c ) → L [ L [ x o y ] ≈ L [ x o y ] ] | |
| 18 | 82 lemma12 L x y = |
| 83 let open ≈-Reasoning ( L ) in | |
| 84 begin L [ x o y ] ∎ | |
| 11 | 85 |
| 4 | 86 open Kleisli |
| 22 | 87 -- η(b) ○ f = f |
| 88 Lemma7 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → | |
| 56 | 89 { T : Functor A A } |
| 21 | 90 ( η : NTrans A A identityFunctor T ) |
| 7 | 91 { μ : NTrans A A (T ○ T) T } |
| 21 | 92 { a : Obj A } ( b : Obj A ) |
| 93 ( f : Hom A a ( FObj T b) ) | |
| 22 | 94 ( m : Monad A T η μ ) |
| 56 | 95 { k : Kleisli A T η μ m} |
| 30 | 96 → A [ join k b (TMap η b) f ≈ f ] |
| 56 | 97 Lemma7 c {T} η b f m {k} = |
| 21 | 98 let open ≈-Reasoning (c) |
| 99 μ = mu ( monad k ) | |
| 100 in | |
| 101 begin | |
| 30 | 102 join k b (TMap η b) f |
| 21 | 103 ≈⟨ refl-hom ⟩ |
| 30 | 104 c [ TMap μ b o c [ FMap T ((TMap η b)) o f ] ] |
| 21 | 105 ≈⟨ IsCategory.associative (Category.isCategory c) ⟩ |
| 30 | 106 c [ c [ TMap μ b o FMap T ((TMap η b)) ] o f ] |
| 28 | 107 ≈⟨ car ( IsMonad.unity2 ( isMonad ( monad k )) ) ⟩ |
| 22 | 108 c [ id (FObj T b) o f ] |
| 21 | 109 ≈⟨ IsCategory.identityL (Category.isCategory c) ⟩ |
| 110 f | |
| 111 ∎ | |
| 7 | 112 |
| 22 | 113 -- f ○ η(a) = f |
| 114 Lemma8 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) | |
| 115 ( T : Functor A A ) | |
| 116 ( η : NTrans A A identityFunctor T ) | |
| 7 | 117 { μ : NTrans A A (T ○ T) T } |
| 22 | 118 ( a : Obj A ) ( b : Obj A ) |
| 119 ( f : Hom A a ( FObj T b) ) | |
| 120 ( m : Monad A T η μ ) | |
| 121 ( k : Kleisli A T η μ m) | |
| 30 | 122 → A [ join k b f (TMap η a) ≈ f ] |
| 22 | 123 Lemma8 c T η a b f m k = |
| 124 begin | |
| 30 | 125 join k b f (TMap η a) |
| 22 | 126 ≈⟨ refl-hom ⟩ |
| 30 | 127 c [ TMap μ b o c [ FMap T f o (TMap η a) ] ] |
| 28 | 128 ≈⟨ cdr ( IsNTrans.naturality ( isNTrans η )) ⟩ |
| 30 | 129 c [ TMap μ b o c [ (TMap η ( FObj T b)) o f ] ] |
| 22 | 130 ≈⟨ IsCategory.associative (Category.isCategory c) ⟩ |
| 30 | 131 c [ c [ TMap μ b o (TMap η ( FObj T b)) ] o f ] |
| 28 | 132 ≈⟨ car ( IsMonad.unity1 ( isMonad ( monad k )) ) ⟩ |
| 22 | 133 c [ id (FObj T b) o f ] |
| 134 ≈⟨ IsCategory.identityL (Category.isCategory c) ⟩ | |
| 135 f | |
| 136 ∎ where | |
| 137 open ≈-Reasoning (c) | |
| 138 μ = mu ( monad k ) | |
| 5 | 139 |
| 22 | 140 -- h ○ (g ○ f) = (h ○ g) ○ f |
| 23 | 141 Lemma9 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) |
| 56 | 142 { T : Functor A A } |
| 143 { η : NTrans A A identityFunctor T } | |
| 144 { μ : NTrans A A (T ○ T) T } | |
| 145 { a b c d : Obj A } | |
| 23 | 146 ( f : Hom A a ( FObj T b) ) |
| 147 ( g : Hom A b ( FObj T c) ) | |
| 148 ( h : Hom A c ( FObj T d) ) | |
| 149 ( m : Monad A T η μ ) | |
| 56 | 150 { k : Kleisli A T η μ m} |
| 22 | 151 → A [ join k d h (join k c g f) ≈ join k d ( join k d h g) f ] |
| 56 | 152 Lemma9 A {T} {η} {μ} {a} {b} {c} {d} f g h m {k} = |
| 24 | 153 begin |
| 23 | 154 join k d h (join k c g f) |
| 30 | 155 ≈⟨⟩ |
| 156 join k d h ( ( TMap μ c o ( FMap T g o f ) ) ) | |
| 28 | 157 ≈⟨ refl-hom ⟩ |
| 30 | 158 ( TMap μ d o ( FMap T h o ( TMap μ c o ( FMap T g o f ) ) ) ) |
| 28 | 159 ≈⟨ cdr ( cdr ( assoc )) ⟩ |
| 30 | 160 ( TMap μ d o ( FMap T h o ( ( TMap μ c o FMap T g ) o f ) ) ) |
| 28 | 161 ≈⟨ assoc ⟩ --- ( f o ( g o h ) ) = ( ( f o g ) o h ) |
| 30 | 162 ( ( TMap μ d o FMap T h ) o ( (TMap μ c o FMap T g ) o f ) ) |
| 25 | 163 ≈⟨ assoc ⟩ |
| 30 | 164 ( ( ( TMap μ d o FMap T h ) o (TMap μ c o FMap T g ) ) o f ) |
| 28 | 165 ≈⟨ car (sym assoc) ⟩ |
| 30 | 166 ( ( TMap μ d o ( FMap T h o ( TMap μ c o FMap T g ) ) ) o f ) |
| 28 | 167 ≈⟨ car ( cdr (assoc) ) ⟩ |
| 30 | 168 ( ( TMap μ d o ( ( FMap T h o TMap μ c ) o FMap T g ) ) o f ) |
| 28 | 169 ≈⟨ car assoc ⟩ |
| 30 | 170 ( ( ( TMap μ d o ( FMap T h o TMap μ c ) ) o FMap T g ) o f ) |
| 28 | 171 ≈⟨ car (car ( cdr ( begin |
| 30 | 172 ( FMap T h o TMap μ c ) |
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ad62c87659ef
join association finish
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
changeset
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173 ≈⟨ nat A μ ⟩ |
| 30 | 174 ( TMap μ (FObj T d) o FMap T (FMap T h) ) |
| 25 | 175 ∎ |
| 176 ))) ⟩ | |
| 30 | 177 ( ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) ) o FMap T g ) o f ) |
| 28 | 178 ≈⟨ car (sym assoc) ⟩ |
| 30 | 179 ( ( TMap μ d o ( ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) o FMap T g ) ) o f ) |
| 28 | 180 ≈⟨ car ( cdr (sym assoc) ) ⟩ |
| 30 | 181 ( ( TMap μ d o ( TMap μ ( FObj T d) o ( FMap T ( FMap T h ) o FMap T g ) ) ) o f ) |
| 28 | 182 ≈⟨ car ( cdr (cdr (sym (distr T )))) ⟩ |
| 30 | 183 ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 28 | 184 ≈⟨ car assoc ⟩ |
| 30 | 185 ( ( ( TMap μ d o TMap μ ( FObj T d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 186 ≈⟨ car ( car ( |
| 27 | 187 begin |
| 30 | 188 ( TMap μ d o TMap μ (FObj T d) ) |
| 27 | 189 ≈⟨ IsMonad.assoc ( isMonad m) ⟩ |
| 30 | 190 ( TMap μ d o FMap T (TMap μ d) ) |
| 27 | 191 ∎ |
| 192 )) ⟩ | |
| 30 | 193 ( ( ( TMap μ d o FMap T ( TMap μ d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 194 ≈⟨ car (sym assoc) ⟩ |
| 30 | 195 ( ( TMap μ d o ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 24 | 196 ≈⟨ sym assoc ⟩ |
| 30 | 197 ( TMap μ d o ( ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) o f ) ) |
| 28 | 198 ≈⟨ cdr ( car ( sym ( distr T ))) ⟩ |
| 30 | 199 ( TMap μ d o ( FMap T ( ( ( TMap μ d ) o ( FMap T h o g ) ) ) o f ) ) |
| 23 | 200 ≈⟨ refl-hom ⟩ |
| 30 | 201 join k d ( ( TMap μ d o ( FMap T h o g ) ) ) f |
| 23 | 202 ≈⟨ refl-hom ⟩ |
| 203 join k d ( join k d h g) f | |
| 24 | 204 ∎ where open ≈-Reasoning (A) |
| 3 | 205 |
| 56 | 206 -- Kleisli-join : {!!} |
| 207 -- Kleisli-join = {!!} | |
| 3 | 208 |
| 56 | 209 -- Kleisli-id : {!!} |
| 210 -- Kleisli-id = {!!} | |
| 211 | |
| 212 -- Lemma10 : {!!} | |
| 213 -- Lemma10 = {!!} | |
| 214 | |
| 215 -- open import Relation.Binary.Core | |
| 216 | |
| 217 -- isKleisliCategory : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) | |
| 218 -- { T : Functor A A } | |
| 219 -- { η : NTrans A A identityFunctor T } | |
| 220 -- { μ : NTrans A A (T ○ T) T } | |
| 221 -- ( m : Monad A T η μ ) | |
| 222 -- { k : Kleisli A T η μ m} -> | |
| 223 -- IsCategory ( Obj A ) ( Hom A ) ( Category._≈_ A ) ( Kleisli-join ) Kleisli-id | |
| 224 -- isKleisliCategory A {T} {η} m = record { isEquivalence = IsCategory.isEquivalence ( Category.isCategory A ) | |
| 225 -- ; identityL = {!!} | |
| 226 -- ; identityR = {!!} | |
| 227 -- ; o-resp-≈ = {!!} | |
| 228 -- ; associative = {!!} | |
| 229 -- } | |
| 230 -- where | |
| 231 -- KidL : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) { T : Functor A A } | |
| 232 -- { η : NTrans A A identityFunctor T } | |
| 233 -- { μ : NTrans A A (T ○ T) T } ( m : Monad A T η μ ) -> {!!} | |
| 234 -- KidL = {!!} | |
| 235 -- KidR : {!!} | |
| 236 -- KidR = {!!} | |
| 237 -- Ko-resp : {!!} | |
| 238 -- Ko-resp = {!!} | |
| 239 -- Kassoc : {!!} | |
| 240 -- Kassoc = {!!} | |
| 3 | 241 |
| 242 -- Kleisli : | |
| 243 -- Kleisli = record { Hom = | |
| 244 -- ; Hom = _⟶_ | |
| 245 -- ; Id = IdProd | |
| 246 -- ; _o_ = _∘_ | |
| 247 -- ; _≈_ = _≈_ | |
| 248 -- ; isCategory = record { isEquivalence = record { refl = λ {φ} → ≈-refl {φ = φ} | |
| 249 -- ; sym = λ {φ ψ} → ≈-symm {φ = φ} {ψ} | |
| 250 -- ; trans = λ {φ ψ σ} → ≈-trans {φ = φ} {ψ} {σ} | |
| 251 -- } | |
| 252 -- ; identityL = identityL | |
| 253 -- ; identityR = identityR | |
| 254 -- ; o-resp-≈ = o-resp-≈ | |
| 255 -- ; associative = associative | |
| 256 -- } | |
| 257 -- } | |
| 56 | 258 |
| 259 open Adjunction | |
| 260 | |
| 261 -- ( \b -> FMap U ( TMap ε ( FObj F b)) ) | |
| 262 Adj2Monad : {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') | |
| 263 { U : Functor B A } | |
| 264 { F : Functor A B } | |
| 265 { η : NTrans A A identityFunctor ( U ○ F ) } | |
| 266 { ε : NTrans B B ( F ○ U ) identityFunctor } → | |
| 267 Adjunction A B U F η ε → Monad A (U ○ F) η {!!} | |
| 268 Adj2Monad A B {U} {F} {η} {ε} adj = record { | |
| 269 isMonad = record { | |
| 270 assoc = {!!}; | |
| 271 unity1 = {!!}; | |
| 272 unity2 = {!!} | |
| 273 } | |
| 274 } |