cat/src/Cat/Category/Monad.agda

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