465 lines
17 KiB
Agda
465 lines
17 KiB
Agda
{-# OPTIONS --cubical --allow-unsolved-metas #-}
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module Cat.Category.Monad where
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open import Agda.Primitive
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open import Data.Product
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open import Cubical
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open import Cat.Category hiding (propIsAssociative)
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open import Cat.Category.Functor as F
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open import Cat.Category.NaturalTransformation
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open import Cat.Categories.Fun
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-- "A monad in the monoidal form" [voe]
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module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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open Category ℂ using (Object ; Arrow ; 𝟙 ; _∘_)
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open NaturalTransformation ℂ ℂ
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record RawMonad : Set ℓ where
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field
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-- TODO rename fields here
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-- R ~ m
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R : EndoFunctor ℂ
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-- η ~ pure
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ηNatTrans : NaturalTransformation F.identity R
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-- μ ~ join
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μNatTrans : NaturalTransformation F[ R ∘ R ] R
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η : Transformation F.identity R
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η = proj₁ ηNatTrans
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ηNat : Natural F.identity R η
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ηNat = proj₂ ηNatTrans
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μ : Transformation F[ R ∘ R ] R
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μ = proj₁ μNatTrans
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μNat : Natural F[ R ∘ R ] R μ
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μNat = proj₂ μNatTrans
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private
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module R = Functor R
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IsAssociative : Set _
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IsAssociative = {X : Object}
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→ μ X ∘ R.func→ (μ X) ≡ μ X ∘ μ (R.func* X)
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IsInverse : Set _
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IsInverse = {X : Object}
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→ μ X ∘ η (R.func* X) ≡ 𝟙
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× μ X ∘ R.func→ (η X) ≡ 𝟙
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IsNatural = ∀ {X Y} f → μ Y ∘ R.func→ f ∘ η X ≡ f
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IsDistributive = ∀ {X Y Z} (g : Arrow Y (R.func* Z)) (f : Arrow X (R.func* Y))
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→ μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f)
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≡ μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f)
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record IsMonad (raw : RawMonad) : Set ℓ where
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open RawMonad raw public
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field
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isAssociative : IsAssociative
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isInverse : IsInverse
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private
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module R = Functor R
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module ℂ = Category ℂ
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isNatural : IsNatural
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isNatural {X} {Y} f = begin
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μ Y ∘ R.func→ f ∘ η X ≡⟨ sym ℂ.isAssociative ⟩
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μ Y ∘ (R.func→ f ∘ η X) ≡⟨ cong (λ φ → μ Y ∘ φ) (sym (ηNat f)) ⟩
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μ Y ∘ (η (R.func* Y) ∘ f) ≡⟨ ℂ.isAssociative ⟩
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μ Y ∘ η (R.func* Y) ∘ f ≡⟨ cong (λ φ → φ ∘ f) (proj₁ isInverse) ⟩
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𝟙 ∘ f ≡⟨ proj₂ ℂ.isIdentity ⟩
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f ∎
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isDistributive : IsDistributive
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isDistributive {X} {Y} {Z} g f = sym done
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where
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module R² = Functor F[ R ∘ R ]
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postulate
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distrib : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
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→ R.func→ (a ∘ b ∘ c)
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≡ R.func→ a ∘ R.func→ b ∘ R.func→ c
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comm : ∀ {A B C D E}
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→ {a : Arrow D E} {b : Arrow C D} {c : Arrow B C} {d : Arrow A B}
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→ a ∘ (b ∘ c ∘ d) ≡ a ∘ b ∘ c ∘ d
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lemmm : μ Z ∘ R.func→ (μ Z) ≡ μ Z ∘ μ (R.func* Z)
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lemmm = isAssociative
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lem4 : μ (R.func* Z) ∘ R².func→ g ≡ R.func→ g ∘ μ Y
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lem4 = μNat g
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done = begin
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μ Z ∘ R.func→ (μ Z ∘ R.func→ g ∘ f) ≡⟨ cong (λ φ → μ Z ∘ φ) distrib ⟩
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μ Z ∘ (R.func→ (μ Z) ∘ R.func→ (R.func→ g) ∘ R.func→ f) ≡⟨⟩
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μ Z ∘ (R.func→ (μ Z) ∘ R².func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver?
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(μ Z ∘ R.func→ (μ Z)) ∘ (R².func→ g ∘ R.func→ f) ≡⟨ cong (λ φ → φ ∘ (R².func→ g ∘ R.func→ f)) lemmm ⟩
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(μ Z ∘ μ (R.func* Z)) ∘ (R².func→ g ∘ R.func→ f) ≡⟨ {!!} ⟩ -- ●-solver?
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μ Z ∘ μ (R.func* Z) ∘ R².func→ g ∘ R.func→ f ≡⟨ {!!} ⟩ -- ●-solver + lem4
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μ Z ∘ R.func→ g ∘ μ Y ∘ R.func→ f ≡⟨ sym (Category.isAssociative ℂ) ⟩
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μ Z ∘ R.func→ g ∘ (μ Y ∘ R.func→ f) ∎
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record Monad : Set ℓ where
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field
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raw : RawMonad
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isMonad : IsMonad raw
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open IsMonad isMonad public
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private
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module _ {m : RawMonad} where
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open RawMonad m
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propIsAssociative : isProp IsAssociative
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propIsAssociative x y i {X}
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= Category.arrowsAreSets ℂ _ _ (x {X}) (y {X}) i
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propIsInverse : isProp IsInverse
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propIsInverse x y i {X} = e1 i , e2 i
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where
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xX = x {X}
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yX = y {X}
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e1 = Category.arrowsAreSets ℂ _ _ (proj₁ xX) (proj₁ yX)
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e2 = Category.arrowsAreSets ℂ _ _ (proj₂ xX) (proj₂ yX)
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open IsMonad
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propIsMonad : (raw : _) → isProp (IsMonad raw)
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IsMonad.isAssociative (propIsMonad raw a b i) j
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= propIsAssociative {raw}
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(isAssociative a) (isAssociative b) i j
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IsMonad.isInverse (propIsMonad raw a b i)
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= propIsInverse {raw}
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(isInverse a) (isInverse b) i
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module _ {m n : Monad} (eq : Monad.raw m ≡ Monad.raw n) where
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open import Cubical.NType.Properties
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eqIsMonad : (λ i → IsMonad (eq i)) [ Monad.isMonad m ≡ Monad.isMonad n ]
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eqIsMonad = lemPropF propIsMonad eq
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Monad≡ : m ≡ n
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Monad.raw (Monad≡ i) = eq i
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Monad.isMonad (Monad≡ i) = eqIsMonad i
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-- "A monad in the Kleisli form" [voe]
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module Kleisli {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Arrow ; 𝟙 ; Object ; _∘_ ; _>>>_)
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-- | Data for a monad.
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--
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-- Note that (>>=) is not expressible in a general category because objects
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-- are not generally types.
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record RawMonad : Set ℓ where
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field
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RR : Object → Object
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-- Note name-change from [voe]
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pure : {X : Object} → ℂ [ X , RR X ]
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bind : {X Y : Object} → ℂ [ X , RR Y ] → ℂ [ RR X , RR Y ]
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-- | functor map
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--
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-- This should perhaps be defined in a "Klesli-version" of functors as well?
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fmap : ∀ {A B} → ℂ [ A , B ] → ℂ [ RR A , RR B ]
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fmap f = bind (pure ∘ f)
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-- | Composition of monads aka. the kleisli-arrow.
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_>=>_ : {A B C : Object} → ℂ [ A , RR B ] → ℂ [ B , RR C ] → ℂ [ A , RR C ]
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f >=> g = f >>> (bind g)
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-- | Flattening nested monads.
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join : {A : Object} → ℂ [ RR (RR A) , RR A ]
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join = bind 𝟙
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------------------
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-- * Monad laws --
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------------------
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-- There may be better names than what I've chosen here.
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IsIdentity = {X : Object}
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→ bind pure ≡ 𝟙 {RR X}
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IsNatural = {X Y : Object} (f : ℂ [ X , RR Y ])
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→ pure >>> (bind f) ≡ f
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IsDistributive = {X Y Z : Object} (g : ℂ [ Y , RR Z ]) (f : ℂ [ X , RR Y ])
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→ (bind f) >>> (bind g) ≡ bind (f >=> g)
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-- | Functor map fusion.
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--
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-- This is really a functor law. Should we have a kleisli-representation of
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-- functors as well and make them a super-class?
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Fusion = {X Y Z : Object} {g : ℂ [ Y , Z ]} {f : ℂ [ X , Y ]}
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→ fmap (g ∘ f) ≡ fmap g ∘ fmap f
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-- In the ("foreign") formulation of a monad `IsNatural`'s analogue here would be:
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IsNaturalForeign : Set _
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IsNaturalForeign = {X : Object} → join {X} ∘ fmap join ≡ join ∘ join
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IsInverse : Set _
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IsInverse = {X : Object} → join {X} ∘ pure ≡ 𝟙 × join {X} ∘ fmap pure ≡ 𝟙
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record IsMonad (raw : RawMonad) : Set ℓ where
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open RawMonad raw public
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field
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isIdentity : IsIdentity
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isNatural : IsNatural
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isDistributive : IsDistributive
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-- | Map fusion is admissable.
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fusion : Fusion
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fusion {g = g} {f} = begin
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fmap (g ∘ f) ≡⟨⟩
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bind ((f >>> g) >>> pure) ≡⟨ cong bind isAssociative ⟩
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bind (f >>> (g >>> pure)) ≡⟨ cong (λ φ → bind (f >>> φ)) (sym (isNatural _)) ⟩
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bind (f >>> (pure >>> (bind (g >>> pure)))) ≡⟨⟩
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bind (f >>> (pure >>> fmap g)) ≡⟨⟩
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bind ((fmap g ∘ pure) ∘ f) ≡⟨ cong bind (sym isAssociative) ⟩
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bind (fmap g ∘ (pure ∘ f)) ≡⟨ sym lem ⟩
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bind (pure ∘ g) ∘ bind (pure ∘ f) ≡⟨⟩
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fmap g ∘ fmap f ∎
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where
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open Category ℂ using (isAssociative)
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lem : fmap g ∘ fmap f ≡ bind (fmap g ∘ (pure ∘ f))
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lem = isDistributive (pure ∘ g) (pure ∘ f)
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-- | This formulation gives rise to the following endo-functor.
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private
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rawR : RawFunctor ℂ ℂ
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RawFunctor.func* rawR = RR
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RawFunctor.func→ rawR = fmap
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isFunctorR : IsFunctor ℂ ℂ rawR
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IsFunctor.isIdentity isFunctorR = begin
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bind (pure ∘ 𝟙) ≡⟨ cong bind (proj₁ ℂ.isIdentity) ⟩
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bind pure ≡⟨ isIdentity ⟩
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𝟙 ∎
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IsFunctor.isDistributive isFunctorR {f = f} {g} = begin
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bind (pure ∘ (g ∘ f)) ≡⟨⟩
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fmap (g ∘ f) ≡⟨ fusion ⟩
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fmap g ∘ fmap f ≡⟨⟩
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bind (pure ∘ g) ∘ bind (pure ∘ f) ∎
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-- TODO: Naming!
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R : EndoFunctor ℂ
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Functor.raw R = rawR
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Functor.isFunctor R = isFunctorR
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private
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open NaturalTransformation ℂ ℂ
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R⁰ : EndoFunctor ℂ
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R⁰ = F.identity
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R² : EndoFunctor ℂ
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R² = F[ R ∘ R ]
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module R = Functor R
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module R⁰ = Functor R⁰
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module R² = Functor R²
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ηTrans : Transformation R⁰ R
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ηTrans A = pure
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ηNatural : Natural R⁰ R ηTrans
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ηNatural {A} {B} f = begin
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ηTrans B ∘ R⁰.func→ f ≡⟨⟩
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pure ∘ f ≡⟨ sym (isNatural _) ⟩
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bind (pure ∘ f) ∘ pure ≡⟨⟩
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fmap f ∘ pure ≡⟨⟩
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R.func→ f ∘ ηTrans A ∎
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μTrans : Transformation R² R
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μTrans C = join
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μNatural : Natural R² R μTrans
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μNatural f = begin
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join ∘ R².func→ f ≡⟨⟩
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bind 𝟙 ∘ R².func→ f ≡⟨⟩
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R².func→ f >>> bind 𝟙 ≡⟨⟩
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fmap (fmap f) >>> bind 𝟙 ≡⟨⟩
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fmap (bind (f >>> pure)) >>> bind 𝟙 ≡⟨⟩
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bind (bind (f >>> pure) >>> pure) >>> bind 𝟙
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≡⟨ isDistributive _ _ ⟩
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bind ((bind (f >>> pure) >>> pure) >=> 𝟙)
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≡⟨⟩
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bind ((bind (f >>> pure) >>> pure) >>> bind 𝟙)
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≡⟨ cong bind ℂ.isAssociative ⟩
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bind (bind (f >>> pure) >>> (pure >>> bind 𝟙))
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≡⟨ cong (λ φ → bind (bind (f >>> pure) >>> φ)) (isNatural _) ⟩
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bind (bind (f >>> pure) >>> 𝟙)
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≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
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bind (bind (f >>> pure))
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≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
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bind (𝟙 >>> bind (f >>> pure)) ≡⟨⟩
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bind (𝟙 >=> (f >>> pure))
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≡⟨ sym (isDistributive _ _) ⟩
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bind 𝟙 >>> bind (f >>> pure) ≡⟨⟩
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bind 𝟙 >>> fmap f ≡⟨⟩
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bind 𝟙 >>> R.func→ f ≡⟨⟩
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R.func→ f ∘ bind 𝟙 ≡⟨⟩
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R.func→ f ∘ join ∎
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where
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ηNatTrans : NaturalTransformation R⁰ R
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proj₁ ηNatTrans = ηTrans
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proj₂ ηNatTrans = ηNatural
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μNatTrans : NaturalTransformation R² R
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proj₁ μNatTrans = μTrans
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proj₂ μNatTrans = μNatural
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isNaturalForeign : IsNaturalForeign
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isNaturalForeign = begin
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fmap join >>> join ≡⟨⟩
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bind (join >>> pure) >>> bind 𝟙
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≡⟨ isDistributive _ _ ⟩
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bind ((join >>> pure) >>> bind 𝟙)
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≡⟨ cong bind ℂ.isAssociative ⟩
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bind (join >>> (pure >>> bind 𝟙))
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≡⟨ cong (λ φ → bind (join >>> φ)) (isNatural _) ⟩
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bind (join >>> 𝟙)
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≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
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bind join ≡⟨⟩
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bind (bind 𝟙)
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≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
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bind (𝟙 >>> bind 𝟙) ≡⟨⟩
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bind (𝟙 >=> 𝟙) ≡⟨ sym (isDistributive _ _) ⟩
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bind 𝟙 >>> bind 𝟙 ≡⟨⟩
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join >>> join ∎
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isInverse : IsInverse
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isInverse = inv-l , inv-r
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where
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inv-l = begin
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pure >>> join ≡⟨⟩
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pure >>> bind 𝟙 ≡⟨ isNatural _ ⟩
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𝟙 ∎
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inv-r = begin
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fmap pure >>> join ≡⟨⟩
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bind (pure >>> pure) >>> bind 𝟙
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≡⟨ isDistributive _ _ ⟩
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bind ((pure >>> pure) >=> 𝟙) ≡⟨⟩
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bind ((pure >>> pure) >>> bind 𝟙)
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≡⟨ cong bind ℂ.isAssociative ⟩
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bind (pure >>> (pure >>> bind 𝟙))
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≡⟨ cong (λ φ → bind (pure >>> φ)) (isNatural _) ⟩
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bind (pure >>> 𝟙)
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≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
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bind pure ≡⟨ isIdentity ⟩
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𝟙 ∎
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record Monad : Set ℓ where
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field
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raw : RawMonad
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isMonad : IsMonad raw
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open IsMonad isMonad public
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postulate propIsMonad : ∀ {raw} → isProp (IsMonad raw)
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Monad≡ : {m n : Monad} → Monad.raw m ≡ Monad.raw n → m ≡ n
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Monad.raw (Monad≡ eq i) = eq i
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Monad.isMonad (Monad≡ {m} {n} eq i) = res i
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where
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-- TODO: PathJ nightmare + `propIsMonad`.
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res : (λ i → IsMonad (eq i)) [ Monad.isMonad m ≡ Monad.isMonad n ]
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res = {!!}
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-- | The monoidal- and kleisli presentation of monads are equivalent.
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--
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-- This is problem 2.3 in [voe].
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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open Category ℂ using (Object ; Arrow ; 𝟙 ; _∘_)
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open Functor using (func* ; func→)
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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-- Note similarity with locally defined things in Kleisly.RawMonad!!
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module _ (m : M.RawMonad) where
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private
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open M.RawMonad m
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module Kraw = K.RawMonad
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RR : Object → Object
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RR = func* R
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pure : {X : Object} → ℂ [ X , RR X ]
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pure {X} = η X
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bind : {X Y : Object} → ℂ [ X , RR Y ] → ℂ [ RR X , RR Y ]
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bind {X} {Y} f = μ Y ∘ func→ R f
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forthRaw : K.RawMonad
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Kraw.RR forthRaw = RR
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Kraw.pure forthRaw = pure
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Kraw.bind forthRaw = bind
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module _ {raw : M.RawMonad} (m : M.IsMonad raw) where
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private
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module MI = M.IsMonad m
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module KI = K.IsMonad
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forthIsMonad : K.IsMonad (forthRaw raw)
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KI.isIdentity forthIsMonad = proj₂ MI.isInverse
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KI.isNatural forthIsMonad = MI.isNatural
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KI.isDistributive forthIsMonad = MI.isDistributive
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forth : M.Monad → K.Monad
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Kleisli.Monad.raw (forth m) = forthRaw (M.Monad.raw m)
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Kleisli.Monad.isMonad (forth m) = forthIsMonad (M.Monad.isMonad m)
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module _ (m : K.Monad) where
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private
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open K.Monad m
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module MR = M.RawMonad
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module MI = M.IsMonad
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backRaw : M.RawMonad
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MR.R backRaw = R
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MR.ηNatTrans backRaw = ηNatTrans
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MR.μNatTrans backRaw = μNatTrans
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private
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open MR backRaw
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module R = Functor (MR.R backRaw)
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backIsMonad : M.IsMonad backRaw
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MI.isAssociative backIsMonad {X} = begin
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μ X ∘ R.func→ (μ X) ≡⟨⟩
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join ∘ fmap (μ X) ≡⟨⟩
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join ∘ fmap join ≡⟨ isNaturalForeign ⟩
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join ∘ join ≡⟨⟩
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μ X ∘ μ (R.func* X) ∎
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MI.isInverse backIsMonad {X} = inv-l , inv-r
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where
|
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inv-l = begin
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μ X ∘ η (R.func* X) ≡⟨⟩
|
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join ∘ pure ≡⟨ proj₁ isInverse ⟩
|
||
𝟙 ∎
|
||
inv-r = begin
|
||
μ X ∘ R.func→ (η X) ≡⟨⟩
|
||
join ∘ fmap pure ≡⟨ proj₂ isInverse ⟩
|
||
𝟙 ∎
|
||
|
||
back : K.Monad → M.Monad
|
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Monoidal.Monad.raw (back m) = backRaw m
|
||
Monoidal.Monad.isMonad (back m) = backIsMonad m
|
||
|
||
-- I believe all the proofs here should be `refl`.
|
||
module _ (m : K.Monad) where
|
||
open K.RawMonad (K.Monad.raw m)
|
||
forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
|
||
K.RawMonad.RR (forthRawEq _) = RR
|
||
K.RawMonad.pure (forthRawEq _) = pure
|
||
-- stuck
|
||
K.RawMonad.bind (forthRawEq i) = {!!}
|
||
|
||
fortheq : (m : K.Monad) → forth (back m) ≡ m
|
||
fortheq m = K.Monad≡ (forthRawEq m)
|
||
|
||
module _ (m : M.Monad) where
|
||
open M.RawMonad (M.Monad.raw m)
|
||
backRawEq : backRaw (forth m) ≡ M.Monad.raw m
|
||
-- stuck
|
||
M.RawMonad.R (backRawEq i) = {!!}
|
||
M.RawMonad.ηNatTrans (backRawEq i) = {!!}
|
||
M.RawMonad.μNatTrans (backRawEq i) = {!!}
|
||
|
||
backeq : (m : M.Monad) → back (forth m) ≡ m
|
||
backeq m = M.Monad≡ (backRawEq m)
|
||
|
||
open import Cubical.GradLemma
|
||
eqv : isEquiv M.Monad K.Monad forth
|
||
eqv = gradLemma forth back fortheq backeq
|
||
|
||
Monoidal≃Kleisli : M.Monad ≃ K.Monad
|
||
Monoidal≃Kleisli = forth , eqv
|