Mercurial > hg > Members > kono > Proof > category
annotate monoid-monad.agda @ 149:2f68a9e0167b
clean up
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Thu, 15 Aug 2013 12:04:50 +0900 |
| parents | 6e80e1aaa8b9 |
| children | 5dc6f3f43507 |
| rev | line source |
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| 129 | 1 open import Category -- https://github.com/konn/category-agda |
| 130 | 2 open import Algebra |
| 129 | 3 open import Level |
| 149 | 4 open import Category.Sets |
| 5 module monoid-monad {c : Level} where | |
| 130 | 6 |
| 142 | 7 open import Algebra.Structures |
| 129 | 8 open import HomReasoning |
| 9 open import cat-utility | |
| 10 open import Category.Cat | |
| 138 | 11 open import Data.Product |
| 12 open import Relation.Binary.Core | |
| 13 open import Relation.Binary | |
| 131 | 14 |
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15 -- open Monoid |
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16 open import Algebra.FunctionProperties using (Op₁; Op₂) |
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17 |
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18 |
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19 record ≡-Monoid c : Set (suc c) where |
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20 infixl 7 _∙_ |
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21 field |
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22 Carrier : Set c |
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23 _∙_ : Op₂ Carrier |
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24 ε : Carrier |
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25 isMonoid : IsMonoid _≡_ _∙_ ε |
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26 open IsMonoid isMonoid public |
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27 |
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28 postulate Mono : ≡-Monoid c |
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29 open ≡-Monoid |
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30 |
| 149 | 31 A = Sets {c} |
| 138 | 32 |
| 139 | 33 -- T : A → (M x A) |
| 134 | 34 |
| 149 | 35 T : Functor A A |
| 138 | 36 T = record { |
| 149 | 37 FObj = λ a → (Carrier Mono) × a |
| 38 ; FMap = λ f → map ( λ x → x ) f | |
| 138 | 39 ; isFunctor = record { |
| 40 identity = IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory Sets )) | |
| 41 ; distr = (IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory Sets ))) | |
| 42 ; ≈-cong = cong1 | |
| 43 } | |
| 44 } where | |
| 139 | 45 cong1 : {ℓ′ : Level} → {a b : Set ℓ′} { f g : Hom (Sets {ℓ′}) a b} → |
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46 Sets [ f ≈ g ] → Sets [ map (λ (x : Carrier Mono) → x) f ≈ map (λ (x : Carrier Mono) → x) g ] |
| 138 | 47 cong1 _≡_.refl = _≡_.refl |
| 129 | 48 |
| 144 | 49 open Functor |
| 50 | |
| 149 | 51 Lemma-MM1 : {a b : Obj A} {f : Hom A a b} → |
| 52 A [ A [ FMap T f o (λ x → ε Mono , x) ] ≈ | |
| 53 A [ (λ x → ε Mono , x) o f ] ] | |
| 54 Lemma-MM1 {a} {b} {f} = let open ≈-Reasoning A renaming ( _o_ to _*_ ) in | |
| 139 | 55 begin |
| 144 | 56 FMap T f o (λ x → ε Mono , x) |
| 139 | 57 ≈⟨⟩ |
| 58 (λ x → ε Mono , x) o f | |
| 59 ∎ | |
| 60 | |
| 61 -- a → (ε,a) | |
| 149 | 62 η : NTrans A A identityFunctor T |
| 139 | 63 η = record { |
| 149 | 64 TMap = λ a → λ(x : a) → ( ε Mono , x ) ; |
| 139 | 65 isNTrans = record { |
| 149 | 66 commute = Lemma-MM1 |
| 139 | 67 } |
| 68 } | |
| 69 | |
| 70 -- (m,(m',a)) → (m*m,a) | |
| 71 | |
| 149 | 72 muMap : (a : Obj A ) → FObj T ( FObj T a ) → Σ (Carrier Mono) (λ x → a ) |
| 139 | 73 muMap a ( m , ( m' , x ) ) = ( _∙_ Mono m m' , x ) |
| 74 | |
| 149 | 75 Lemma-MM2 : {a b : Obj A} {f : Hom A a b} → |
| 76 A [ A [ FMap T f o (λ x → muMap a x) ] ≈ | |
| 77 A [ (λ x → muMap b x) o FMap (T ○ T) f ] ] | |
| 78 Lemma-MM2 {a} {b} {f} = let open ≈-Reasoning A renaming ( _o_ to _*_ ) in | |
| 139 | 79 begin |
| 80 FMap T f o (λ x → muMap a x) | |
| 81 ≈⟨⟩ | |
| 82 (λ x → muMap b x) o FMap (T ○ T) f | |
| 83 ∎ | |
| 84 | |
| 149 | 85 μ : NTrans A A ( T ○ T ) T |
| 139 | 86 μ = record { |
| 149 | 87 TMap = λ a → λ x → muMap a x ; |
| 139 | 88 isNTrans = record { |
| 149 | 89 commute = λ{a} {b} {f} → Lemma-MM2 {a} {b} {f} |
| 139 | 90 } |
| 91 } | |
| 141 | 92 |
| 93 open NTrans | |
| 94 | |
| 144 | 95 Lemma-MM33 : {a : Level} {b : Level} {A : Set a} {B : A → Set b} {x : Σ A B } → (((proj₁ x) , proj₂ x ) ≡ x ) |
| 142 | 96 Lemma-MM33 = _≡_.refl |
| 97 | |
| 149 | 98 Lemma-MM34 : ∀{ x : Carrier Mono } → ( (Mono ∙ ε Mono) x ≡ x ) |
| 147 | 99 Lemma-MM34 {x} = (( proj₁ ( IsMonoid.identity ( isMonoid Mono )) ) x ) |
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100 |
| 149 | 101 Lemma-MM35 : ∀{ x : Carrier Mono } → ((Mono ∙ x) (ε Mono)) ≡ x |
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102 Lemma-MM35 {x} = ( proj₂ ( IsMonoid.identity ( isMonoid Mono )) ) x |
| 141 | 103 |
| 149 | 104 Lemma-MM36 : ∀{ x y z : Carrier Mono } → ((Mono ∙ (Mono ∙ x) y) z) ≡ (_∙_ Mono x (_∙_ Mono y z ) ) |
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105 Lemma-MM36 {x} {y} {z} = ( IsMonoid.assoc ( isMonoid Mono )) x y z |
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106 |
| 149 | 107 -- Functional Extensionarity (We cannot prove this in Agda / Coq, just assumming ) |
| 108 postulate Extensionarity : {f g : Carrier Mono → Carrier Mono } → (∀ {x} → (f x ≡ g x)) → ( f ≡ g ) | |
| 144 | 109 |
| 149 | 110 postulate Extensionarity3 : {f g : Carrier Mono → Carrier Mono → Carrier Mono → Carrier Mono } → |
| 111 (∀{x y z} → f x y z ≡ g x y z ) → ( f ≡ g ) | |
| 144 | 112 |
| 149 | 113 Lemma-MM9 : ( λ(x : Carrier Mono) → ( Mono ∙ ε Mono ) x ) ≡ ( λ(x : Carrier Mono) → x ) |
| 114 Lemma-MM9 = Extensionarity Lemma-MM34 | |
| 144 | 115 |
| 149 | 116 Lemma-MM10 : ( λ x → ((Mono ∙ x) (ε Mono))) ≡ ( λ x → x ) |
| 117 Lemma-MM10 = Extensionarity Lemma-MM35 | |
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118 |
| 149 | 119 Lemma-MM11 : (λ x y z → ((Mono ∙ (Mono ∙ x) y) z)) ≡ (λ x y z → ( _∙_ Mono x (_∙_ Mono y z ) )) |
| 120 Lemma-MM11 = Extensionarity3 Lemma-MM36 | |
| 145 | 121 |
| 149 | 122 MonoidMonad : Monad A T η μ |
| 141 | 123 MonoidMonad = record { |
| 124 isMonad = record { | |
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125 unity1 = Lemma-MM3 ; |
| 141 | 126 unity2 = Lemma-MM4 ; |
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127 assoc = Lemma-MM5 |
| 141 | 128 } |
| 129 } where | |
| 147 | 130 Lemma-MM3 : {a : Obj A} → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 149 | 131 Lemma-MM3 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in |
| 141 | 132 begin |
| 133 TMap μ a o TMap η ( FObj T a ) | |
| 134 ≈⟨⟩ | |
| 135 ( λ x → ((Mono ∙ ε Mono) (proj₁ x) , proj₂ x )) | |
| 149 | 136 ≈⟨ cong ( λ f → ( λ x → ( ( f (proj₁ x) ) , proj₂ x ))) ( Lemma-MM9 ) ⟩ |
| 147 | 137 ( λ (x : FObj T a) → (proj₁ x) , proj₂ x ) |
| 144 | 138 ≈⟨⟩ |
| 141 | 139 ( λ x → x ) |
| 140 ≈⟨⟩ | |
| 141 Id {_} {_} {_} {A} (FObj T a) | |
| 142 ∎ | |
| 143 Lemma-MM4 : {a : Obj A} → A [ A [ TMap μ a o (FMap T (TMap η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ] | |
| 149 | 144 Lemma-MM4 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in |
| 144 | 145 begin |
| 146 TMap μ a o (FMap T (TMap η a )) | |
| 147 ≈⟨⟩ | |
| 148 ( λ x → (Mono ∙ proj₁ x) (ε Mono) , proj₂ x ) | |
| 149 | 149 ≈⟨ cong ( λ f → ( λ x → ( f (proj₁ x) ) , proj₂ x )) ( Lemma-MM10 ) ⟩ |
| 144 | 150 ( λ x → (proj₁ x) , proj₂ x ) |
| 151 ≈⟨⟩ | |
| 152 ( λ x → x ) | |
| 153 ≈⟨⟩ | |
| 154 Id {_} {_} {_} {A} (FObj T a) | |
| 155 ∎ | |
| 141 | 156 Lemma-MM5 : {a : Obj A} → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] |
| 149 | 157 Lemma-MM5 {a} = let open ≈-Reasoning (A) renaming ( _o_ to _*_ ) in |
| 144 | 158 begin |
| 159 TMap μ a o TMap μ ( FObj T a ) | |
| 160 ≈⟨⟩ | |
| 161 ( λ x → (Mono ∙ (Mono ∙ proj₁ x) (proj₁ (proj₂ x))) (proj₁ (proj₂ (proj₂ x))) , proj₂ (proj₂ (proj₂ x))) | |
| 149 | 162 ≈⟨ cong ( λ f → ( λ x → |
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163 (( f ( proj₁ x ) ((proj₁ (proj₂ x))) ((proj₁ (proj₂ (proj₂ x))) )) , proj₂ (proj₂ (proj₂ x)) ) |
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164 )) Lemma-MM11 ⟩ |
| 144 | 165 ( λ x → (Mono ∙ proj₁ x) ((Mono ∙ proj₁ (proj₂ x)) (proj₁ (proj₂ (proj₂ x)))) , proj₂ (proj₂ (proj₂ x))) |
| 166 ≈⟨⟩ | |
| 167 TMap μ a o FMap T (TMap μ a) | |
| 168 ∎ | |
| 141 | 169 |
| 170 | |
| 171 |