254 lines
8.7 KiB
Agda
254 lines
8.7 KiB
Agda
{---
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The Kleisli formulation of monads
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---}
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{-# OPTIONS --cubical --allow-unsolved-metas #-}
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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 Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
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open import Cubical.GradLemma using (gradLemma)
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open import Cat.Category
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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 Kleisli form" [voe]
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module Cat.Category.Monad.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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omap : Object → Object
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pure : {X : Object} → ℂ [ X , omap X ]
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bind : {X Y : Object} → ℂ [ X , omap Y ] → ℂ [ omap X , omap 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 [ omap A , omap 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 , omap B ] → ℂ [ B , omap C ] → ℂ [ A , omap 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 [ omap (omap A) , omap 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 {omap X}
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IsNatural = {X Y : Object} (f : ℂ [ X , omap Y ])
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→ pure >>> (bind f) ≡ f
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IsDistributive = {X Y Z : Object} (g : ℂ [ Y , omap Z ]) (f : ℂ [ X , omap 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 ℂ.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 ℂ.isAssociative) ⟩
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bind (fmap g ∘ (pure ∘ f)) ≡⟨ sym distrib ⟩
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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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distrib : fmap g ∘ fmap f ≡ bind (fmap g ∘ (pure ∘ f))
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distrib = 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.omap rawR = omap
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RawFunctor.fmap 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 (ℂ.rightIdentity) ⟩
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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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-- FIXME 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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pureT : Transformation R⁰ R
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pureT A = pure
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pureN : Natural R⁰ R pureT
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pureN {A} {B} f = begin
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pureT B ∘ R⁰.fmap 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.fmap f ∘ pureT A ∎
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joinT : Transformation R² R
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joinT C = join
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joinN : Natural R² R joinT
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joinN f = begin
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join ∘ R².fmap f ≡⟨⟩
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bind 𝟙 ∘ R².fmap f ≡⟨⟩
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R².fmap 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 ℂ.leftIdentity ⟩
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bind (bind (f >>> pure))
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≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
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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.fmap f ≡⟨⟩
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R.fmap f ∘ bind 𝟙 ≡⟨⟩
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R.fmap f ∘ join ∎
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pureNT : NaturalTransformation R⁰ R
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proj₁ pureNT = pureT
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proj₂ pureNT = pureN
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joinNT : NaturalTransformation R² R
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proj₁ joinNT = joinT
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proj₂ joinNT = joinN
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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 ℂ.leftIdentity ⟩
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bind join ≡⟨⟩
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bind (bind 𝟙)
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≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
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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 ℂ.leftIdentity ⟩
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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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private
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module _ (raw : RawMonad) where
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open RawMonad raw
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propIsIdentity : isProp IsIdentity
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propIsIdentity x y i = ℂ.arrowsAreSets _ _ x y i
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propIsNatural : isProp IsNatural
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propIsNatural x y i = λ f
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→ ℂ.arrowsAreSets _ _ (x f) (y f) i
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propIsDistributive : isProp IsDistributive
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propIsDistributive x y i = λ g f
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→ ℂ.arrowsAreSets _ _ (x g f) (y g f) i
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open IsMonad
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propIsMonad : (raw : _) → isProp (IsMonad raw)
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IsMonad.isIdentity (propIsMonad raw x y i)
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= propIsIdentity raw (isIdentity x) (isIdentity y) i
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IsMonad.isNatural (propIsMonad raw x y i)
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= propIsNatural raw (isNatural x) (isNatural y) i
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IsMonad.isDistributive (propIsMonad raw x y i)
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= propIsDistributive raw (isDistributive x) (isDistributive y) i
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module _ {m n : Monad} (eq : Monad.raw m ≡ Monad.raw n) where
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private
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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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