Use long name

This commit is contained in:
Frederik Hanghøj Iversen 2018-05-11 13:20:03 +02:00
parent 30cf0bb765
commit 4d73514ab5
4 changed files with 162 additions and 142 deletions

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@ -10,8 +10,8 @@ open import Cat.Category.NaturalTransformation
open import Cat.Category.Yoneda
open import Cat.Category.Monoid
open import Cat.Category.Monad
open Cat.Category.Monad.Monoidal
open Cat.Category.Monad.Kleisli
open import Cat.Category.Monad.Monoidal
open import Cat.Category.Monad.Kleisli
open import Cat.Category.Monad.Voevodsky
open import Cat.Categories.Sets

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@ -28,186 +28,205 @@ import Cat.Category.Monad.Monoidal
import Cat.Category.Monad.Kleisli
open import Cat.Categories.Fun
module Monoidal = Cat.Category.Monad.Monoidal
module Kleisli = Cat.Category.Monad.Kleisli
-- | The monoidal- and kleisli presentation of monads are equivalent.
module _ {a b : Level} ( : Category a b) where
open Cat.Category.NaturalTransformation using (NaturalTransformation ; propIsNatural)
private
module = Category
open using (Object ; Arrow ; identity ; _<<<_ ; _>>>_)
module M = Monoidal
module K = Kleisli
module _ (m : M.RawMonad) where
open M.RawMonad m
module Monoidal = Cat.Category.Monad.Monoidal
module Kleisli = Cat.Category.Monad.Kleisli
forthRaw : K.RawMonad
K.RawMonad.omap forthRaw = Romap
K.RawMonad.pure forthRaw = pureT _
K.RawMonad.bind forthRaw = bind
module _ (m : Monoidal.RawMonad) where
open Monoidal.RawMonad m
module _ {raw : M.RawMonad} (m : M.IsMonad raw) where
private
module MI = M.IsMonad m
forthIsMonad : K.IsMonad (forthRaw raw)
K.IsMonad.isIdentity forthIsMonad = snd MI.isInverse
K.IsMonad.isNatural forthIsMonad = MI.isNatural
K.IsMonad.isDistributive forthIsMonad = MI.isDistributive
toKleisliRaw : Kleisli.RawMonad
Kleisli.RawMonad.omap toKleisliRaw = Romap
Kleisli.RawMonad.pure toKleisliRaw = pure
Kleisli.RawMonad.bind toKleisliRaw = bind
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 _ {raw : Monoidal.RawMonad} (m : Monoidal.IsMonad raw) where
open Monoidal.IsMonad m
module _ (m : K.Monad) where
open K.Monad m
open Kleisli.RawMonad (toKleisliRaw raw) using (_>=>_)
toKleisliIsMonad : Kleisli.IsMonad (toKleisliRaw raw)
Kleisli.IsMonad.isIdentity toKleisliIsMonad = begin
bind pure ≡⟨⟩
join <<< (fmap pure) ≡⟨ snd isInverse
identity
Kleisli.IsMonad.isNatural toKleisliIsMonad f = begin
pure >=> f ≡⟨⟩
pure >>> bind f ≡⟨⟩
bind f <<< pure ≡⟨⟩
(join <<< fmap f) <<< pure ≡⟨ isNatural f
f
Kleisli.IsMonad.isDistributive toKleisliIsMonad f g = begin
bind g >>> bind f ≡⟨⟩
(join <<< fmap g) >>> (join <<< fmap f) ≡⟨ isDistributive f g
bind (g >=> f)
-- Kleisli.IsMonad.isDistributive toKleisliIsMonad = isDistributive
backRaw : M.RawMonad
M.RawMonad.R backRaw = R
M.RawMonad.pureNT backRaw = pureNT
M.RawMonad.joinNT backRaw = joinNT
toKleisli : Monoidal.Monad Kleisli.Monad
Kleisli.Monad.raw (toKleisli m) = toKleisliRaw (Monoidal.Monad.raw m)
Kleisli.Monad.isMonad (toKleisli m) = toKleisliIsMonad (Monoidal.Monad.isMonad m)
private
open M.RawMonad backRaw renaming
( join to join*
; pure to pure*
; bind to bind*
; fmap to fmap*
)
module R = Functor (M.RawMonad.R backRaw)
module _ (m : Kleisli.Monad) where
open Kleisli.Monad m
backIsMonad : M.IsMonad backRaw
M.IsMonad.isAssociative backIsMonad = begin
join* <<< R.fmap join* ≡⟨⟩
toMonoidalRaw : Monoidal.RawMonad
Monoidal.RawMonad.R toMonoidalRaw = R
Monoidal.RawMonad.pureNT toMonoidalRaw = pureNT
Monoidal.RawMonad.joinNT toMonoidalRaw = joinNT
open Monoidal.RawMonad toMonoidalRaw renaming
( join to join*
; pure to pure*
; bind to bind*
; fmap to fmap*
) using ()
toMonoidalIsMonad : Monoidal.IsMonad toMonoidalRaw
Monoidal.IsMonad.isAssociative toMonoidalIsMonad = begin
join* <<< fmap join* ≡⟨⟩
join <<< fmap join ≡⟨ isNaturalForeign
join <<< join
M.IsMonad.isInverse backIsMonad {X} = inv-l , inv-r
Monoidal.IsMonad.isInverse toMonoidalIsMonad {X} = inv-l , inv-r
where
inv-l = begin
join <<< pure ≡⟨ fst isInverse
identity
inv-r = begin
joinT X <<< R.fmap (pureT X) ≡⟨⟩
join* <<< fmap* pure* ≡⟨⟩
join <<< fmap pure ≡⟨ snd isInverse
identity
back : K.Monad M.Monad
Monoidal.Monad.raw (back m) = backRaw m
Monoidal.Monad.isMonad (back m) = backIsMonad m
toMonoidal : Kleisli.Monad Monoidal.Monad
Monoidal.Monad.raw (toMonoidal m) = toMonoidalRaw m
Monoidal.Monad.isMonad (toMonoidal m) = toMonoidalIsMonad m
module _ (m : K.Monad) where
module _ (m : Kleisli.Monad) where
private
open K.Monad m
open Kleisli.Monad 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 join <<< fmap f) ≡⟨⟩
(λ f bind (f >>> pure) >>> bind identity) ≡⟨ funExt lem
(λ f bind f) ≡⟨⟩
bind
Kleisli.RawMonad.bind (toKleisliRaw (toMonoidalRaw m)) {X} {Y}
bind
bindEq {X} {Y} = funExt lem
where
lem : (f : Arrow X (omap Y))
bind (f >>> pure) >>> bind identity
bind f
lem f = begin
join <<< fmap f
≡⟨⟩
bind (f >>> pure) >>> bind identity
≡⟨ isDistributive _ _
bind ((f >>> pure) >=> identity)
≡⟨⟩
bind ((f >>> pure) >>> bind identity)
≡⟨ cong bind .isAssociative
bind (f >>> (pure >>> bind identity))
≡⟨⟩
bind (f >>> (pure >=> identity))
≡⟨ cong (λ φ bind (f >>> φ)) (isNatural _)
bind (f >>> identity)
≡⟨ cong bind .leftIdentity
bind f
forthRawEq : forthRaw (backRaw m) K.Monad.raw m
K.RawMonad.omap (forthRawEq _) = omap
K.RawMonad.pure (forthRawEq _) = pure
K.RawMonad.bind (forthRawEq i) = bindEq i
toKleisliRawEq : toKleisliRaw (toMonoidalRaw m) Kleisli.Monad.raw m
Kleisli.RawMonad.omap (toKleisliRawEq i) = (begin
Kleisli.RawMonad.omap (toKleisliRaw (toMonoidalRaw m)) ≡⟨⟩
Monoidal.RawMonad.Romap (toMonoidalRaw m) ≡⟨⟩
omap
) i
Kleisli.RawMonad.pure (toKleisliRawEq i) = (begin
Kleisli.RawMonad.pure (toKleisliRaw (toMonoidalRaw m)) ≡⟨⟩
Monoidal.RawMonad.pure (toMonoidalRaw m) ≡⟨⟩
pure
) i
Kleisli.RawMonad.bind (toKleisliRawEq i) = bindEq i
fortheq : (m : K.Monad) forth (back m) m
fortheq m = K.Monad≡ (forthRawEq m)
toKleislieq : (m : Kleisli.Monad) toKleisli (toMonoidal m) m
toKleislieq m = Kleisli.Monad≡ (toKleisliRawEq m)
module _ (m : M.Monad) where
module _ (m : Monoidal.Monad) where
private
open M.Monad m
module KM = K.Monad (forth m)
open Monoidal.Monad m
-- module KM = Kleisli.Monad (toKleisli m)
open Kleisli.Monad (toKleisli m) renaming
( bind to bind* ; omap to omap* ; join to join*
; fmap to fmap* ; pure to pure* ; R to R*)
using ()
module R = Functor R
omapEq : KM.omap Romap
omapEq : omap* Romap
omapEq = refl
bindEq : {X Y} {f : Arrow X (Romap Y)} KM.bind f bind f
bindEq : {X Y} {f : Arrow X (Romap Y)} bind* f bind f
bindEq {X} {Y} {f} = begin
KM.bind f ≡⟨⟩
joinT Y <<< fmap f ≡⟨⟩
bind f
bind* f ≡⟨⟩
join <<< fmap f ≡⟨⟩
bind f
joinEq : {X} KM.join joinT X
joinEq : {X} join* joinT X
joinEq {X} = begin
KM.join ≡⟨⟩
KM.bind identity ≡⟨⟩
bind identity ≡⟨⟩
joinT X <<< fmap identity ≡⟨ cong (λ φ _ <<< φ) R.isIdentity
joinT X <<< identity ≡⟨ .rightIdentity
joinT X
join* ≡⟨⟩
bind* identity ≡⟨⟩
bind identity ≡⟨⟩
join <<< fmap identity ≡⟨ cong (λ φ _ <<< φ) R.isIdentity
join <<< identity ≡⟨ .rightIdentity
join
fmapEq : {A B} KM.fmap {A} {B} fmap
fmapEq : {A B} fmap* {A} {B} fmap
fmapEq {A} {B} = funExt (λ f begin
KM.fmap f ≡⟨⟩
KM.bind (f >>> KM.pure) ≡⟨⟩
bind (f >>> pureT _) ≡⟨⟩
fmap (f >>> pureT B) >>> joinT B ≡⟨⟩
fmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ φ >>> joinT B) R.isDistributive
fmap f >>> fmap (pureT B) >>> joinT B ≡⟨ .isAssociative
joinT B <<< fmap (pureT B) <<< fmap f ≡⟨ cong (λ φ φ <<< fmap f) (snd isInverse)
identity <<< fmap f ≡⟨ .leftIdentity
fmap f
fmap* f ≡⟨⟩
bind* (f >>> pure*) ≡⟨⟩
bind (f >>> pure) ≡⟨⟩
fmap (f >>> pure) >>> join ≡⟨⟩
fmap (f >>> pure) >>> join ≡⟨ cong (λ φ φ >>> joinT B) R.isDistributive
fmap f >>> fmap pure >>> join ≡⟨ .isAssociative
join <<< fmap pure <<< fmap f ≡⟨ cong (λ φ φ <<< fmap f) (snd isInverse)
identity <<< fmap f ≡⟨ .leftIdentity
fmap f
)
rawEq : Functor.raw KM.R Functor.raw R
rawEq : Functor.raw R* Functor.raw R
RawFunctor.omap (rawEq i) = omapEq i
RawFunctor.fmap (rawEq i) = fmapEq i
Req : M.RawMonad.R (backRaw (forth m)) R
Req : Monoidal.RawMonad.R (toMonoidalRaw (toKleisli m)) R
Req = Functor≡ rawEq
pureTEq : M.RawMonad.pureT (backRaw (forth m)) pureT
pureTEq : Monoidal.RawMonad.pureT (toMonoidalRaw (toKleisli m)) pureT
pureTEq = funExt (λ X refl)
pureNTEq : (λ i NaturalTransformation Functors.identity (Req i))
[ M.RawMonad.pureNT (backRaw (forth m)) pureNT ]
[ Monoidal.RawMonad.pureNT (toMonoidalRaw (toKleisli m)) pureNT ]
pureNTEq = lemSigP (λ i propIsNatural Functors.identity (Req i)) _ _ pureTEq
joinTEq : M.RawMonad.joinT (backRaw (forth m)) joinT
joinTEq : Monoidal.RawMonad.joinT (toMonoidalRaw (toKleisli m)) joinT
joinTEq = funExt (λ X begin
M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
KM.join ≡⟨⟩
joinT X <<< fmap identity ≡⟨ cong (λ φ joinT X <<< φ) R.isIdentity
joinT X <<< identity ≡⟨ .rightIdentity
joinT X )
Monoidal.RawMonad.joinT (toMonoidalRaw (toKleisli m)) X ≡⟨⟩
join* ≡⟨⟩
join <<< fmap identity ≡⟨ cong (λ φ join <<< φ) R.isIdentity
join <<< identity ≡⟨ .rightIdentity
join )
joinNTEq : (λ i NaturalTransformation F[ Req i Req i ] (Req i))
[ M.RawMonad.joinNT (backRaw (forth m)) joinNT ]
[ Monoidal.RawMonad.joinNT (toMonoidalRaw (toKleisli m)) joinNT ]
joinNTEq = lemSigP (λ i propIsNatural F[ Req i Req i ] (Req i)) _ _ joinTEq
backRawEq : backRaw (forth m) M.Monad.raw m
M.RawMonad.R (backRawEq i) = Req i
M.RawMonad.pureNT (backRawEq i) = pureNTEq i
M.RawMonad.joinNT (backRawEq i) = joinNTEq i
toMonoidalRawEq : toMonoidalRaw (toKleisli m) Monoidal.Monad.raw m
Monoidal.RawMonad.R (toMonoidalRawEq i) = Req i
Monoidal.RawMonad.pureNT (toMonoidalRawEq i) = pureNTEq i
Monoidal.RawMonad.joinNT (toMonoidalRawEq i) = joinNTEq i
backeq : (m : M.Monad) back (forth m) m
backeq m = M.Monad≡ (backRawEq m)
eqv : isEquiv M.Monad K.Monad forth
eqv = gradLemma forth back fortheq backeq
toMonoidaleq : (m : Monoidal.Monad) toMonoidal (toKleisli m) m
toMonoidaleq m = Monoidal.Monad≡ (toMonoidalRawEq m)
open import Cat.Equivalence
Monoidal≊Kleisli : M.Monad K.Monad
Monoidal≊Kleisli = forth , back , funExt backeq , funExt fortheq
Monoidal≊Kleisli : Monoidal.Monad Kleisli.Monad
Monoidal≊Kleisli = toKleisli , toMonoidal , funExt toMonoidaleq , funExt toKleislieq
Monoidal≡Kleisli : M.Monad K.Monad
Monoidal≡Kleisli : Monoidal.Monad Kleisli.Monad
Monoidal≡Kleisli = isoToPath Monoidal≊Kleisli

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@ -18,7 +18,7 @@ private
open Category using (Object ; Arrow ; identity ; _<<<_)
open import Cat.Category.NaturalTransformation
using (NaturalTransformation ; Transformation ; Natural)
using (NaturalTransformation ; Transformation ; Natural ; NaturalTransformation≡)
record RawMonad : Set where
field
@ -78,15 +78,39 @@ record IsMonad (raw : RawMonad) : Set where
isNatural : IsNatural
isNatural {X} {Y} f = begin
joinT Y <<< R.fmap f <<< pureT X ≡⟨ sym .isAssociative
joinT Y <<< (R.fmap f <<< pureT X) ≡⟨ cong (λ φ joinT Y <<< φ) (sym (pureN f))
joinT Y <<< (pureT (R.omap Y) <<< f) ≡⟨ .isAssociative
joinT Y <<< pureT (R.omap Y) <<< f ≡⟨ cong (λ φ φ <<< f) (fst isInverse)
identity <<< f ≡⟨ .leftIdentity
f
join <<< fmap f <<< pure ≡⟨ sym .isAssociative
join <<< (fmap f <<< pure) ≡⟨ cong (λ φ join <<< φ) (sym (pureN f))
join <<< (pure <<< f) ≡⟨ .isAssociative
join <<< pure <<< f ≡⟨ cong (λ φ φ <<< f) (fst isInverse)
identity <<< f ≡⟨ .leftIdentity
f
isDistributive : IsDistributive
isDistributive {X} {Y} {Z} g f = sym aux
isDistributive {X} {Y} {Z} g f = begin
join <<< fmap g <<< (join <<< fmap f)
≡⟨ Category.isAssociative
join <<< fmap g <<< join <<< fmap f
≡⟨ cong (_<<< fmap f) (sym .isAssociative)
(join <<< (fmap g <<< join)) <<< fmap f
≡⟨ cong (λ φ φ <<< fmap f) (cong (_<<<_ join) (sym (joinN g)))
(join <<< (join <<< R².fmap g)) <<< fmap f
≡⟨ cong (_<<< fmap f) .isAssociative
((join <<< join) <<< R².fmap g) <<< fmap f
≡⟨⟩
join <<< join <<< R².fmap g <<< fmap f
≡⟨ sym .isAssociative
(join <<< join) <<< (R².fmap g <<< fmap f)
≡⟨ cong (λ φ φ <<< (R².fmap g <<< fmap f)) (sym isAssociative)
(join <<< fmap join) <<< (R².fmap g <<< fmap f)
≡⟨ sym .isAssociative
join <<< (fmap join <<< (R².fmap g <<< fmap f))
≡⟨ cong (_<<<_ join) .isAssociative
join <<< (fmap join <<< R².fmap g <<< fmap f)
≡⟨⟩
join <<< (fmap join <<< fmap (fmap g) <<< fmap f)
≡⟨ cong (λ φ join <<< φ) (sym distrib3)
join <<< fmap (join <<< fmap g <<< f)
where
module R² = Functor F[ R R ]
distrib3 : {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
@ -96,31 +120,6 @@ record IsMonad (raw : RawMonad) : Set where
R.fmap (a <<< b <<< c) ≡⟨ R.isDistributive
R.fmap (a <<< b) <<< R.fmap c ≡⟨ cong (_<<< _) R.isDistributive
R.fmap a <<< R.fmap b <<< R.fmap c
aux = begin
joinT Z <<< R.fmap (joinT Z <<< R.fmap g <<< f)
≡⟨ cong (λ φ joinT Z <<< φ) distrib3
joinT Z <<< (R.fmap (joinT Z) <<< R.fmap (R.fmap g) <<< R.fmap f)
≡⟨⟩
joinT Z <<< (R.fmap (joinT Z) <<< R².fmap g <<< R.fmap f)
≡⟨ cong (_<<<_ (joinT Z)) (sym .isAssociative)
joinT Z <<< (R.fmap (joinT Z) <<< (R².fmap g <<< R.fmap f))
≡⟨ .isAssociative
(joinT Z <<< R.fmap (joinT Z)) <<< (R².fmap g <<< R.fmap f)
≡⟨ cong (λ φ φ <<< (R².fmap g <<< R.fmap f)) isAssociative
(joinT Z <<< joinT (R.omap Z)) <<< (R².fmap g <<< R.fmap f)
≡⟨ .isAssociative
joinT Z <<< joinT (R.omap Z) <<< R².fmap g <<< R.fmap f
≡⟨⟩
((joinT Z <<< joinT (R.omap Z)) <<< R².fmap g) <<< R.fmap f
≡⟨ cong (_<<< R.fmap f) (sym .isAssociative)
(joinT Z <<< (joinT (R.omap Z) <<< R².fmap g)) <<< R.fmap f
≡⟨ cong (λ φ φ <<< R.fmap f) (cong (_<<<_ (joinT Z)) (joinN g))
(joinT Z <<< (R.fmap g <<< joinT Y)) <<< R.fmap f
≡⟨ cong (_<<< R.fmap f) .isAssociative
joinT Z <<< R.fmap g <<< joinT Y <<< R.fmap f
≡⟨ sym (Category.isAssociative )
joinT Z <<< R.fmap g <<< (joinT Y <<< R.fmap f)
record Monad : Set where
field

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@ -10,6 +10,8 @@ open import Cat.Category
open import Cat.Category.Functor as F
import Cat.Category.NaturalTransformation
open import Cat.Category.Monad
import Cat.Category.Monad.Monoidal as Monoidal
import Cat.Category.Monad.Kleisli as Kleisli
open import Cat.Categories.Fun
open import Cat.Equivalence