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Tidy up

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Frederik Hanghøj Iversen 2018-03-06 10:16:42 +01:00
parent 0cebe1e866
commit 485703c85e
1 changed files with 46 additions and 53 deletions
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99
src/Cat/Category/Monad.agda
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@ -7,6 +7,7 @@ open import Data.Product
open import Cubical
open import Cubical.NType.Properties using (lemPropF ; lemSig)
open import Cubical.GradLemma using (gradLemma)
open import Cat.Category
open import Cat.Category.Functor as F
@ -357,25 +358,27 @@ module Kleisli {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
isMonad : IsMonad raw
open IsMonad isMonad public
module _ (raw : RawMonad) where
open RawMonad raw
propIsIdentity : isProp IsIdentity
propIsIdentity x y i = ℂ.arrowsAreSets _ _ x y i
propIsNatural : isProp IsNatural
propIsNatural x y i = λ f
→ ℂ.arrowsAreSets _ _ (x f) (y f) i
propIsDistributive : isProp IsDistributive
propIsDistributive x y i = λ g f
→ ℂ.arrowsAreSets _ _ (x g f) (y g f) i
private
module _ (raw : RawMonad) where
open RawMonad raw
propIsIdentity : isProp IsIdentity
propIsIdentity x y i = ℂ.arrowsAreSets _ _ x y i
propIsNatural : isProp IsNatural
propIsNatural x y i = λ f
→ ℂ.arrowsAreSets _ _ (x f) (y f) i
propIsDistributive : isProp IsDistributive
propIsDistributive x y i = λ g f
→ ℂ.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
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
private
eqIsMonad : (λ i → IsMonad (eq i)) [ Monad.isMonad m ≡ Monad.isMonad n ]
@ -400,7 +403,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
open M.RawMonad m
forthRaw : K.RawMonad
K.RawMonad.omap forthRaw = Romap
K.RawMonad.omap forthRaw = Romap
K.RawMonad.pure forthRaw = pureT _
K.RawMonad.bind forthRaw = bind
@ -413,63 +416,58 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
K.IsMonad.isDistributive forthIsMonad = MI.isDistributive
forth : M.Monad → K.Monad
Kleisli.Monad.raw (forth m) = forthRaw (M.Monad.raw m)
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
private
open K.Monad m
module MR = M.RawMonad
module MI = M.IsMonad
open K.Monad m
backRaw : M.RawMonad
MR.R backRaw = R
MR.pureNT backRaw = pureNT
MR.joinNT backRaw = joinNT
M.RawMonad.R backRaw = R
M.RawMonad.pureNT backRaw = pureNT
M.RawMonad.joinNT backRaw = joinNT
private
open MR backRaw
module R = Functor (MR.R backRaw)
open M.RawMonad backRaw
module R = Functor (M.RawMonad.R backRaw)
backIsMonad : M.IsMonad backRaw
MI.isAssociative backIsMonad {X} = begin
M.IsMonad.isAssociative backIsMonad {X} = begin
joinT X ∘ R.func→ (joinT X) ≡⟨⟩
join ∘ fmap (joinT X) ≡⟨⟩
join ∘ fmap join ≡⟨ isNaturalForeign ⟩
join ∘ join ≡⟨⟩
join ∘ fmap (joinT X) ≡⟨⟩
join ∘ fmap join ≡⟨ isNaturalForeign ⟩
join ∘ join ≡⟨⟩
joinT X ∘ joinT (R.func* X) ∎
MI.isInverse backIsMonad {X} = inv-l , inv-r
M.IsMonad.isInverse backIsMonad {X} = inv-l , inv-r
where
inv-l = begin
joinT X ∘ pureT (R.func* X) ≡⟨⟩
join ∘ pure ≡⟨ proj₁ isInverse ⟩
𝟙 ∎
join ∘ pure ≡⟨ proj₁ isInverse ⟩
𝟙 ∎
inv-r = begin
joinT X ∘ R.func→ (pureT X) ≡⟨⟩
join ∘ fmap pure ≡⟨ proj₂ isInverse ⟩
𝟙 ∎
join ∘ fmap pure ≡⟨ proj₂ isInverse ⟩
𝟙 ∎
back : K.Monad → M.Monad
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.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 → joinT Y ∘ func→ R f) ≡⟨⟩
(λ f → join ∘ fmap f) ≡⟨⟩
(λ f → bind (f >>> pure) >>> bind 𝟙) ≡⟨ funExt lem ⟩
(λ f → bind f) ≡⟨⟩
bind ∎
(λ f → join ∘ fmap f) ≡⟨⟩
(λ f → bind (f >>> pure) >>> bind 𝟙) ≡⟨ funExt lem ⟩
(λ f → bind f) ≡⟨⟩
bind ∎
where
joinT = proj₁ joinNT
lem : (f : Arrow X (omap Y)) → bind (f >>> pure) >>> bind 𝟙 ≡ bind f
lem : (f : Arrow X (omap Y))
→ bind (f >>> pure) >>> bind 𝟙
≡ bind f
lem f = begin
bind (f >>> pure) >>> bind 𝟙
≡⟨ isDistributive _ _ ⟩
@ -481,13 +479,9 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
bind f ∎
_&_ : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} → A → (A → B) → B
x & f = f x
forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
K.RawMonad.omap (forthRawEq _) = omap
K.RawMonad.pure (forthRawEq _) = pure
-- stuck
K.RawMonad.bind (forthRawEq i) = bindEq i
fortheq : (m : K.Monad) → forth (back m) ≡ m
@ -543,14 +537,13 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
[ M.RawMonad.pureNT (backRaw (forth m)) ≡ pureNT ]
backRawEq : backRaw (forth m) ≡ M.Monad.raw m
-- stuck
M.RawMonad.R (backRawEq i) = Req i
M.RawMonad.R (backRawEq i) = Req i
M.RawMonad.pureNT (backRawEq i) = {!!} -- pureNTEq i
M.RawMonad.joinNT (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