functor-combinators-0.3.3.0: src/Data/HBifunctor/Tensor.hs
-- |
-- Module : Data.HBifunctor.Tensor
-- Copyright : (c) Justin Le 2019
-- License : BSD3
--
-- Maintainer : justin@jle.im
-- Stability : experimental
-- Portability : non-portable
--
-- This module provides tools for working with binary functor combinators.
--
-- "Data.Functor.HFunctor" deals with /single/ functor combinators
-- (transforming a single functor). This module provides tools for working
-- with combinators that combine and mix two functors "together".
--
-- The binary analog of 'HFunctor' is 'HBifunctor': we can map
-- a structure-transforming function over both of the transformed functors.
--
-- 'Tensor' gives some extra properties of your binary functor combinator:
-- associativity and identity (see docs for 'Tensor' for more details).
--
-- The binary analog of 'Interpret' is 'MonoidIn'. If your combinator @t@
-- and target functor @f@ is an instance of @'MonoidIn' t f@, it means you
-- can "interpret" out of your tensored values, and also "generate" values
-- of @f@.
--
-- @
-- 'biretract' :: (f ':+:' f) a -> f a
-- 'pureT' :: 'V1' a -> f a
--
-- biretract :: 'Plus' f => (f ':*:' f) a -> f a
-- pureT :: Plus f => 'Proxy' a -> f a
--
-- biretract :: 'Applicative' f => 'Day' f f a -> f a
-- pureT :: Applicative f => 'Identity' a -> f a
--
-- biretract :: 'Monad' f => 'Comp' f f a -> f a
-- pureT :: Monad f => 'Identity' a -> f a
-- @
--
module Data.HBifunctor.Tensor (
-- * 'Tensor'
Tensor(..)
, rightIdentity
, leftIdentity
, sumLeftIdentity
, sumRightIdentity
, prodLeftIdentity
, prodRightIdentity
-- * 'MonoidIn'
, MonoidIn(..)
, nilLB
, consLB
, unconsLB
, retractLB
, interpretLB
-- ** Utility
, inL
, inR
, outL
, outR
, prodOutL
, prodOutR
, WrapF(..)
, WrapLB(..)
-- * 'Matchable'
, Matchable(..)
, splittingNE
, matchingLB
) where
import Control.Applicative.Free
import Control.Applicative.ListF
import Control.Applicative.Step
import Control.Monad.Freer.Church
import Control.Monad.Trans.Compose
import Control.Natural
import Control.Natural.IsoF
import Data.Data
import Data.Function
import Data.Functor.Apply.Free
import Data.Functor.Bind
import Data.Functor.Classes
import Data.Functor.Contravariant
import Data.Functor.Contravariant.Conclude
import Data.Functor.Contravariant.Decide
import Data.Functor.Contravariant.Divise
import Data.Functor.Contravariant.Divisible
import Data.Functor.Contravariant.Divisible.Free
import Data.Functor.Contravariant.Night (Night(..), Not(..))
import Data.Functor.Day (Day(..))
import Data.Functor.Identity
import Data.Functor.Invariant
import Data.Functor.Plus
import Data.Functor.Product
import Data.Functor.Sum
import Data.Functor.These
import Data.HBifunctor
import Data.HBifunctor.Associative
import Data.HFunctor
import Data.HFunctor.Internal
import Data.HFunctor.Interpret
import Data.Kind
import Data.List.NonEmpty (NonEmpty(..))
import GHC.Generics
import qualified Data.Functor.Contravariant.Coyoneda as CCY
import qualified Data.Functor.Contravariant.Day as CD
import qualified Data.Functor.Contravariant.Night as N
import qualified Data.Functor.Day as D
import qualified Data.Map.NonEmpty as NEM
-- | An 'Associative' 'HBifunctor' can be a 'Tensor' if there is some
-- identity @i@ where @t i f@ and @t f i@ are equivalent to just @f@.
--
-- That is, "enhancing" @f@ with @t i@ does nothing.
--
-- The methods in this class provide us useful ways of navigating
-- a @'Tensor' t@ with respect to this property.
--
-- The 'Tensor' is essentially the 'HBifunctor' equivalent of 'Inject',
-- with 'intro1' and 'intro2' taking the place of 'inject'.
--
-- Formally, we can say that @t@ enriches a the category of
-- endofunctors with monoid strcture: it turns our endofunctor category
-- into a "monoidal category".
--
-- Different instances of @t@ each enrich the endofunctor category in
-- different ways, giving a different monoidal category.
class (Associative t, Inject (ListBy t)) => Tensor t i | t -> i where
-- | The "monoidal functor combinator" induced by @t@.
--
-- A value of type @ListBy t f a@ is /equivalent/ to one of:
--
-- * @I a@ -- zero fs
-- * @f a@ -- one f
-- * @t f f a@ -- two fs
-- * @t f (t f f) a@ -- three fs
-- * @t f (t f (t f f)) a@
-- * @t f (t f (t f (t f f))) a@
-- * .. etc
--
-- For example, for ':*:', we have 'ListF'. This is because:
--
-- @
-- 'Proxy' ~ 'ListF' [] ~ 'nilLB' \@(':*:')
-- x ~ ListF [x] ~ 'inject' x
-- x :*: y ~ ListF [x,y] ~ 'toListBy' (x :*: y)
-- x :*: y :*: z ~ ListF [x,y,z]
-- -- etc.
-- @
--
-- You can create an "empty" one with 'nilLB', a "singleton" one with
-- 'inject', or else one from a single @t f f@ with 'toListBy'.
--
-- See 'Data.HBifunctor.Associative.NonEmptyBy' for a "non-empty"
-- version of this type.
type ListBy t :: (Type -> Type) -> Type -> Type
-- | Because @t f (I t)@ is equivalent to @f@, we can always "insert"
-- @f@ into @t f (I t)@.
--
-- This is analogous to 'inject' from 'Inject', but for 'HBifunctor's.
intro1 :: f ~> t f i
-- | Because @t (I t) g@ is equivalent to @f@, we can always "insert"
-- @g@ into @t (I t) g@.
--
-- This is analogous to 'inject' from 'Inject', but for 'HBifunctor's.
intro2 :: g ~> t i g
-- | Witnesses the property that @i@ is the identity of @t@: @t
-- f i@ always leaves @f@ unchanged, so we can always just drop the
-- @i@.
elim1 :: FunctorBy t f => t f i ~> f
-- | Witnesses the property that @i@ is the identity of @t@: @t i g@
-- always leaves @g@ unchanged, so we can always just drop the @i t@.
elim2 :: FunctorBy t g => t i g ~> g
-- | If a @'ListBy' t f@ represents multiple applications of @t f@ to
-- itself, then we can also "append" two @'ListBy' t f@s applied to
-- themselves into one giant @'ListBy' t f@ containing all of the @t f@s.
--
-- Note that this essentially gives an instance for @'SemigroupIn'
-- t (ListBy t f)@, for any functor @f@; this is witnessed by
-- 'WrapLB'.
appendLB :: t (ListBy t f) (ListBy t f) ~> ListBy t f
-- | Lets you convert an @'NonEmptyBy' t f@ into a single application of @f@ to
-- @'ListBy' t f@.
--
-- Analogous to a function @'Data.List.NonEmpty.NonEmpty' a -> (a,
-- [a])@
--
-- Note that this is not reversible in general unless we have
-- @'Matchable' t@.
splitNE :: NonEmptyBy t f ~> t f (ListBy t f)
-- | An @'ListBy' t f@ is either empty, or a single application of @t@ to @f@
-- and @ListBy t f@ (the "head" and "tail"). This witnesses that
-- isomorphism.
--
-- To /use/ this property, see 'nilLB', 'consLB', and 'unconsLB'.
splittingLB :: ListBy t f <~> i :+: t f (ListBy t f)
-- | Embed a direct application of @f@ to itself into a @'ListBy' t f@.
toListBy :: t f f ~> ListBy t f
toListBy = reviewF (splittingLB @t)
. R1
. hright (inject @(ListBy t))
-- | @'NonEmptyBy' t f@ is "one or more @f@s", and @'ListBy t f@ is "zero or more
-- @f@s". This function lets us convert from one to the other.
--
-- This is analogous to a function @'Data.List.NonEmpty.NonEmpty' a ->
-- [a]@.
--
-- Note that because @t@ is not inferrable from the input or output
-- type, you should call this using /-XTypeApplications/:
--
-- @
-- 'fromNE' \@(':*:') :: 'NonEmptyF' f a -> 'ListF' f a
-- fromNE \@'Comp' :: 'Free1' f a -> 'Free' f a
-- @
fromNE :: NonEmptyBy t f ~> ListBy t f
fromNE = reviewF (splittingLB @t) . R1 . splitNE @t
{-# MINIMAL intro1, intro2, elim1, elim2, appendLB, splitNE, splittingLB #-}
-- | @f@ is isomorphic to @t f i@: that is, @i@ is the identity of @t@, and
-- leaves @f@ unchanged.
rightIdentity :: (Tensor t i, FunctorBy t f) => f <~> t f i
rightIdentity = isoF intro1 elim1
-- | @g@ is isomorphic to @t i g@: that is, @i@ is the identity of @t@, and
-- leaves @g@ unchanged.
leftIdentity :: (Tensor t i, FunctorBy t g) => g <~> t i g
leftIdentity = isoF intro2 elim2
-- | 'leftIdentity' ('intro1' and 'elim1') for ':+:' actually does not
-- require 'Functor'. This is the more general version.
sumLeftIdentity :: f <~> V1 :+: f
sumLeftIdentity = isoF R1 (absurd1 !+! id)
-- | 'rightIdentity' ('intro2' and 'elim2') for ':+:' actually does not
-- require 'Functor'. This is the more general version.
sumRightIdentity :: f <~> f :+: V1
sumRightIdentity = isoF L1 (id !+! absurd1)
-- | 'leftIdentity' ('intro1' and 'elim1') for ':*:' actually does not
-- require 'Functor'. This is the more general version.
prodLeftIdentity :: f <~> Proxy :*: f
prodLeftIdentity = isoF (Proxy :*:) (\case _ :*: y -> y)
-- | 'rightIdentity' ('intro2' and 'elim2') for ':*:' actually does not
-- require 'Functor'. This is the more general version.
prodRightIdentity :: g <~> g :*: Proxy
prodRightIdentity = isoF (:*: Proxy) (\case x :*: _ -> x)
-- | A poly-kinded version of 'outL' for ':*:'.
prodOutL :: f :*: g ~> f
prodOutL (x :*: _) = x
-- | A poly-kinded version of 'outR' for ':*:'.
prodOutR :: f :*: g ~> g
prodOutR (_ :*: y) = y
-- | This class effectively gives us a way to generate a value of @f a@
-- based on an @i a@, for @'Tensor' t i@. Having this ability makes a lot
-- of interesting functions possible when used with 'biretract' from
-- 'SemigroupIn' that weren't possible without it: it gives us a "base
-- case" for recursion in a lot of cases.
--
-- Essentially, we get an @i ~> f@, 'pureT', where we can introduce an @f
-- a@ as long as we have an @i a@.
--
-- Formally, if we have @'Tensor' t i@, we are enriching the category of
-- endofunctors with monoid structure, turning it into a monoidal category.
-- Different choices of @t@ give different monoidal categories.
--
-- A functor @f@ is known as a "monoid in the (monoidal) category
-- of endofunctors on @t@" if we can 'biretract':
--
-- @
-- t f f ~> f
-- @
--
-- and also 'pureT':
--
-- @
-- i ~> f
-- @
--
-- This gives us a few interesting results in category theory, which you
-- can stil reading about if you don't care:
--
-- * /All/ functors are monoids in the monoidal category
-- on ':+:'
-- * The class of functors that are monoids in the monoidal
-- category on ':*:' is exactly the functors that are instances of
-- 'Plus'.
-- * The class of functors that are monoids in the monoidal
-- category on 'Day' is exactly the functors that are instances of
-- 'Applicative'.
-- * The class of functors that are monoids in the monoidal
-- category on 'Comp' is exactly the functors that are instances of
-- 'Monad'.
--
-- This is the meaning behind the common adage, "monads are just monoids
-- in the category of endofunctors". It means that if you enrich the
-- category of endofunctors to be monoidal with 'Comp', then the class
-- of functors that are monoids in that monoidal category are exactly
-- what monads are. However, the adage is a little misleading: there
-- are many other ways to enrich the category of endofunctors to be
-- monoidal, and 'Comp' is just one of them. Similarly, the class of
-- functors that are monoids in the category of endofunctors enriched by
-- 'Day' are 'Applicative'.
--
-- Note that instances of this class are /intended/ to be written with @t@
-- and @i@ to be fixed type constructors, and @f@ to be allowed to vary
-- freely:
--
-- @
-- instance Monad f => MonoidIn Comp Identity f
-- @
--
-- Any other sort of instance and it's easy to run into problems with type
-- inference. If you want to write an instance that's "polymorphic" on
-- tensor choice, use the 'WrapHBF' and 'WrapF' newtype wrappers over type
-- variables, where the third argument also uses a type constructor:
--
-- @
-- instance MonoidIn (WrapHBF t) (WrapF i) (MyFunctor t i)
-- @
--
-- This will prevent problems with overloaded instances.
class (Tensor t i, SemigroupIn t f) => MonoidIn t i f where
-- | If we have an @i@, we can generate an @f@ based on how it
-- interacts with @t@.
--
-- Specialized (and simplified), this type is:
--
-- @
-- 'pureT' \@'Day' :: 'Applicative' f => 'Identity' a -> f a -- 'pure'
-- pureT \@'Comp' :: 'Monad' f => Identity a -> f a -- 'return'
-- pureT \@(':*:') :: 'Plus' f => 'Proxy' a -> f a -- 'zero'
-- @
--
-- Note that because @t@ appears nowhere in the input or output types,
-- you must always use this with explicit type application syntax (like
-- @pureT \@Day@)
--
-- Along with 'biretract', this function makes @f@ a monoid in the
-- category of endofunctors with respect to tensor @t@.
pureT :: i ~> f
default pureT :: Interpret (ListBy t) f => i ~> f
pureT = retract . reviewF (splittingLB @t) . L1
-- | An implementation of 'retract' that works for any instance of
-- @'MonoidIn' t i@ for @'ListBy' t@.
--
-- Can be useful as a default implementation if you already have 'MonoidIn'
-- implemented.
retractLB :: forall t i f. MonoidIn t i f => ListBy t f ~> f
retractLB = (pureT @t !*! biretract @t . hright (retractLB @t))
. unconsLB @t
-- | An implementation of 'interpret' that works for any instance of
-- @'MonoidIn' t i@ for @'ListBy' t@.
--
-- Can be useful as a default implementation if you already have 'MonoidIn'
-- implemented.
interpretLB :: forall t i g f. MonoidIn t i f => (g ~> f) -> ListBy t g ~> f
interpretLB f = retractLB @t . hmap f
-- | Create the "empty 'ListBy'".
--
-- If @'ListBy' t f@ represents multiple applications of @t f@ with
-- itself, then @nilLB@ gives us "zero applications of @f@".
--
-- Note that @t@ cannot be inferred from the input or output type of
-- 'nilLB', so this function must always be called with -XTypeApplications:
--
-- @
-- 'nilLB' \@'Day' :: 'Identity' '~>' 'Ap' f
-- nilLB \@'Comp' :: Identity ~> 'Free' f
-- nilLB \@(':*:') :: 'Proxy' ~> 'ListF' f
-- @
--
-- Note that this essentially gives an instance for @'MonoidIn' t i (ListBy
-- t f)@, for any functor @f@; this is witnessed by 'WrapLB'.
nilLB :: forall t i f. Tensor t i => i ~> ListBy t f
nilLB = reviewF (splittingLB @t) . L1
-- | Lets us "cons" an application of @f@ to the front of an @'ListBy' t f@.
consLB :: Tensor t i => t f (ListBy t f) ~> ListBy t f
consLB = reviewF splittingLB . R1
-- | "Pattern match" on an @'ListBy' t@
--
-- An @'ListBy' t f@ is either empty, or a single application of @t@ to @f@
-- and @ListBy t f@ (the "head" and "tail")
--
-- This is analogous to the function @'Data.List.uncons' :: [a] -> Maybe
-- (a, [a])@.
unconsLB :: Tensor t i => ListBy t f ~> i :+: t f (ListBy t f)
unconsLB = viewF splittingLB
-- | Convenient wrapper over 'intro1' that lets us introduce an arbitrary
-- functor @g@ to the right of an @f@.
--
-- You can think of this as an 'HBifunctor' analogue of 'inject'.
inL
:: forall t i f g. MonoidIn t i g
=> f ~> t f g
inL = hright (pureT @t) . intro1
-- | Convenient wrapper over 'intro2' that lets us introduce an arbitrary
-- functor @f@ to the right of a @g@.
--
-- You can think of this as an 'HBifunctor' analogue of 'inject'.
inR
:: forall t i f g. MonoidIn t i f
=> g ~> t f g
inR = hleft (pureT @t) . intro2
-- | Convenient wrapper over 'elim1' that lets us drop one of the arguments
-- of a 'Tensor' for free, without requiring any extra constraints (like
-- for 'binterpret').
--
-- See 'prodOutL' for a version that does not require @'Functor' f@,
-- specifically for ':*:'.
outL
:: (Tensor t Proxy, FunctorBy t f)
=> t f g ~> f
outL = elim1 . hright absorb
-- | Convenient wrapper over 'elim2' that lets us drop one of the arguments
-- of a 'Tensor' for free, without requiring any constraints (like for
-- 'binterpret').
--
-- See 'prodOutR' for a version that does not require @'Functor' g@,
-- specifically for ':*:'.
outR
:: (Tensor t Proxy, FunctorBy t g)
=> t f g ~> g
outR = elim2 . hleft absorb
-- | For some @t@, we have the ability to "statically analyze" the @'ListBy' t@
-- and pattern match and manipulate the structure without ever
-- interpreting or retracting. These are 'Matchable'.
class Tensor t i => Matchable t i where
-- | The inverse of 'splitNE'. A consing of @f@ to @'ListBy' t f@ is
-- non-empty, so it can be represented as an @'NonEmptyBy' t f@.
--
-- This is analogous to a function @'uncurry' ('Data.List.NonEmpty.:|')
-- :: (a, [a]) -> 'Data.List.NonEmpty.NonEmpty' a@.
unsplitNE :: FunctorBy t f => t f (ListBy t f) ~> NonEmptyBy t f
-- | "Pattern match" on an @'ListBy' t f@: it is either empty, or it is
-- non-empty (and so can be an @'NonEmptyBy' t f@).
--
-- This is analgous to a function @'Data.List.NonEmpty.nonEmpty' :: [a]
-- -> Maybe ('Data.List.NonEmpty.NonEmpty' a)@.
--
-- Note that because @t@ cannot be inferred from the input or output
-- type, you should use this with /-XTypeApplications/:
--
-- @
-- 'matchLB' \@'Day' :: 'Ap' f a -> ('Identity' :+: 'Ap1' f) a
-- @
--
-- Note that you can recursively "unroll" a 'ListBy' completely into
-- a 'Data.HFunctor.Chain.Chain' by using
-- 'Data.HFunctor.Chain.unrollLB'.
matchLB :: FunctorBy t f => ListBy t f ~> i :+: NonEmptyBy t f
-- | An @'NonEmptyBy' t f@ is isomorphic to an @f@ consed with an @'ListBy' t f@, like
-- how a @'Data.List.NonEmpty.NonEmpty' a@ is isomorphic to @(a, [a])@.
splittingNE
:: (Matchable t i, FunctorBy t f)
=> NonEmptyBy t f <~> t f (ListBy t f)
splittingNE = isoF splitNE unsplitNE
-- | An @'ListBy' t f@ is isomorphic to either the empty case (@i@) or the
-- non-empty case (@'NonEmptyBy' t f@), like how @[a]@ is isomorphic to @'Maybe'
-- ('Data.List.NonEmpty.NonEmpty' a)@.
matchingLB
:: forall t i f. (Matchable t i, FunctorBy t f)
=> ListBy t f <~> i :+: NonEmptyBy t f
matchingLB = isoF (matchLB @t) (nilLB @t !*! fromNE @t)
instance Tensor (:*:) Proxy where
type ListBy (:*:) = ListF
intro1 = (:*: Proxy)
intro2 = (Proxy :*:)
elim1 (x :*: ~Proxy) = x
elim2 (~Proxy :*: y ) = y
appendLB (ListF xs :*: ListF ys) = ListF (xs ++ ys)
splitNE = nonEmptyProd
splittingLB = isoF to_ from_
where
to_ = \case
ListF [] -> L1 Proxy
ListF (x:xs) -> R1 (x :*: ListF xs)
from_ = \case
L1 ~Proxy -> ListF []
R1 (x :*: ListF xs) -> ListF (x:xs)
toListBy (x :*: y) = ListF [x, y]
-- | Instances of 'Plus' are monoids in the monoidal category on
-- ':*:'.
instance Plus f => MonoidIn (:*:) Proxy f where
pureT _ = zero
instance Tensor Product Proxy where
type ListBy Product = ListF
intro1 = (`Pair` Proxy)
intro2 = (Proxy `Pair`)
elim1 (Pair x ~Proxy) = x
elim2 (Pair ~Proxy y) = y
appendLB (ListF xs `Pair` ListF ys) = ListF (xs ++ ys)
splitNE = viewF prodProd . nonEmptyProd
splittingLB = isoF to_ from_
where
to_ = \case
ListF [] -> L1 Proxy
ListF (x:xs) -> R1 (x `Pair` ListF xs)
from_ = \case
L1 ~Proxy -> ListF []
R1 (x `Pair` ListF xs) -> ListF (x:xs)
toListBy (Pair x y) = ListF [x, y]
-- | Instances of 'Plus' are monoids in the monoidal category on
-- 'Product'.
instance Plus f => MonoidIn Product Proxy f where
pureT _ = zero
instance Tensor Day Identity where
type ListBy Day = Ap
intro1 = D.intro2
intro2 = D.intro1
elim1 = D.elim2
elim2 = D.elim1
appendLB (Day x y z) = z <$> x <*> y
splitNE = ap1Day
splittingLB = isoF to_ from_
where
to_ = \case
Pure x -> L1 (Identity x)
Ap x xs -> R1 (Day x xs (&))
from_ = \case
L1 (Identity x) -> Pure x
R1 (Day x xs f) -> Ap x (flip f <$> xs)
toListBy (Day x y z) = Ap x (Ap y (Pure (flip z)))
-- | Instances of 'Applicative' are monoids in the monoidal category on
-- the covariant 'Day'.
--
-- Note that because of typeclass constraints, this requires 'Apply' as
-- well as 'Applicative'. But, you can get a "local" instance of 'Apply'
-- for any 'Applicative' using
-- 'Data.Functor.Combinators.Unsafe.unsafeApply'.
instance (Apply f, Applicative f) => MonoidIn Day Identity f where
pureT = generalize
-- | @since 0.3.0.0
instance Tensor CD.Day Proxy where
type ListBy CD.Day = Div
intro1 = CD.intro2
intro2 = CD.intro1
elim1 = CD.day1
elim2 = CD.day2
appendLB (CD.Day x y z) = divide z x y
splitNE = go . splitNE @(:*:) . NonEmptyF . unDiv1
where
go (CCY.Coyoneda f x :*: ListF xs) = CD.Day x (Div xs) (\y -> (f y, y))
splittingLB = isoF (ListF . unDiv) (Div . runListF) . splittingLB @(:*:) . isoF (hright to_) (hright from_)
where
to_ (CCY.Coyoneda f x :*: ListF xs) = CD.Day x (Div xs) (\y -> (f y, y))
from_ (CD.Day x (Div xs) f) = CCY.Coyoneda (fst . f) x :*: contramap (snd . f) (ListF xs)
toListBy (CD.Day x y f) = Div . runListF . toListBy $
CCY.Coyoneda (fst . f) x :*: CCY.Coyoneda (snd . f) y
-- | Instances of 'Divisible' are monoids in the monoidal category on
-- contravariant 'CD.Day'.
--
-- Note that because of typeclass constraints, this requires 'Divise' as
-- well as 'Divisible'. But, you can get a "local" instance of 'Divise'
-- for any 'Divisible' using
-- 'Data.Functor.Combinators.Unsafe.unsafeDivise'.
--
-- @since 0.3.0.0
instance (Divise f, Divisible f) => MonoidIn CD.Day Proxy f where
pureT _ = conquer
-- | @since 0.3.0.0
instance Tensor Night Not where
type ListBy Night = Dec
intro1 = N.intro2
intro2 = N.intro1
elim1 = N.elim2
elim2 = N.elim1
appendLB (Night x y z) = decide z x y
splitNE (Dec1 f x xs) = Night x xs f
splittingLB = isoF to_ from_
where
to_ = \case
Lose f -> L1 (Not f)
Choose f x xs -> R1 (Night x xs f)
from_ = \case
L1 (Not f) -> Lose f
R1 (Night x xs f) -> Choose f x xs
toListBy (Night x y z) = Choose z x (inject y)
-- | Instances of 'Conclude' are monoids in the monoidal category on 'Night'.
instance Conclude f => MonoidIn Night Not f where
pureT (Not x) = conclude x
instance Tensor (:+:) V1 where
type ListBy (:+:) = Step
intro1 = L1
intro2 = R1
elim1 = \case
L1 x -> x
R1 y -> absurd1 y
elim2 = \case
L1 x -> absurd1 x
R1 y -> y
appendLB = id !*! stepUp . R1
splitNE = stepDown
splittingLB = stepping . sumLeftIdentity
toListBy = \case
L1 x -> Step 0 x
R1 x -> Step 1 x
-- | All functors are monoids in the monoidal category on ':+:'.
instance MonoidIn (:+:) V1 f where
pureT = absurd1
instance Tensor Sum V1 where
type ListBy Sum = Step
intro1 = InL
intro2 = InR
elim1 = \case
InL x -> x
InR y -> absurd1 y
elim2 = \case
InL x -> absurd1 x
InR y -> y
appendLB = id !*! stepUp . R1
splitNE = viewF sumSum . stepDown
splittingLB = stepping
. sumLeftIdentity
. overHBifunctor id sumSum
toListBy = \case
InL x -> Step 0 x
InR x -> Step 1 x
-- | All functors are monoids in the monoidal category on 'Sum'.
instance MonoidIn Sum V1 f where
pureT = absurd1
instance Tensor These1 V1 where
type ListBy These1 = Steps
intro1 = This1
intro2 = That1
elim1 = \case
This1 x -> x
That1 y -> absurd1 y
These1 _ y -> absurd1 y
elim2 = \case
This1 x -> absurd1 x
That1 y -> y
These1 x _ -> absurd1 x
appendLB = \case
This1 x -> x
That1 y -> stepsUp . That1 $ y
These1 x y -> x <> y
splitNE = stepsDown . flaggedVal . getComposeT
splittingLB = steppings . sumLeftIdentity
toListBy = \case
This1 x -> Steps $ NEM.singleton 0 x
That1 y -> Steps $ NEM.singleton 1 y
These1 x y -> Steps $ NEM.fromDistinctAscList ((0, x) :| [(1, y)])
instance Alt f => MonoidIn These1 V1 f where
pureT = absurd1
instance Tensor Comp Identity where
type ListBy Comp = Free
intro1 = (:>>= Identity)
intro2 = (Identity () :>>=) . const
elim1 (x :>>= y) = runIdentity . y <$> x
elim2 (x :>>= y) = y (runIdentity x)
appendLB (x :>>= y) = x >>= y
splitNE = free1Comp
splittingLB = isoF to_ from_
where
to_ :: Free f ~> Identity :+: Comp f (Free f)
to_ = foldFree' (L1 . Identity) $ \y n -> R1 $
y :>>= (from_ . n)
from_ :: Identity :+: Comp f (Free f) ~> Free f
from_ = generalize
!*! (\case x :>>= f -> liftFree x >>= f)
toListBy (x :>>= y) = liftFree x >>= (inject . y)
-- | Instances of 'Monad' are monoids in the monoidal category on
-- 'Comp'.
--
-- This instance is the "proof" that "monads are the monoids in the
-- category of endofunctors (enriched with 'Comp')"
--
-- Note that because of typeclass constraints, this requires 'Bind' as
-- well as 'Monad'. But, you can get a "local" instance of 'Apply'
-- for any 'Monad' using
-- 'Data.Functor.Combinators.Unsafe.unsafeBind'.
instance (Bind f, Monad f) => MonoidIn Comp Identity f where
pureT = generalize
instance Matchable (:*:) Proxy where
unsplitNE = ProdNonEmpty
matchLB = fromListF
instance Matchable Product Proxy where
unsplitNE = ProdNonEmpty . reviewF prodProd
matchLB = fromListF
instance Matchable Day Identity where
unsplitNE = DayAp1
matchLB = fromAp
-- | Instances of 'Conclude' are monoids in the monoidal category on 'Night'.
--
-- @since 0.3.0.0
instance Matchable CD.Day Proxy where
unsplitNE (CD.Day x (Div xs) f) = Div1 . runNonEmptyF . unsplitNE $
CCY.Coyoneda (fst . f) x :*: contramap (snd . f) (ListF xs)
matchLB = hright (Div1 . runNonEmptyF) . matchLB @(:*:) . ListF . unDiv
-- | @since 0.3.0.0
instance Matchable Night Not where
unsplitNE (Night x xs f) = Dec1 f x xs
matchLB = \case
Lose f -> L1 (Not f)
Choose f x xs -> R1 (Dec1 f x xs)
instance Matchable (:+:) V1 where
unsplitNE = stepUp
matchLB = R1
instance Matchable Sum V1 where
unsplitNE = stepUp . reviewF sumSum
matchLB = R1
-- We can't write this until we get an isomorphism between MF These1 and SF These1
-- instance Matchable These1 where
-- unsplitNE = stepsUp
-- matchLB = R1
-- | A newtype wrapper meant to be used to define polymorphic 'MonoidIn'
-- instances. See documentation for 'MonoidIn' for more information.
--
-- Please do not ever define an instance of 'MonoidIn' "naked" on the
-- third parameter:
--
-- @
-- instance MonidIn (WrapHBF t) (WrapF i) f
-- @
--
-- As that would globally ruin everything using 'WrapHBF'.
newtype WrapF f a = WrapF { unwrapF :: f a }
deriving (Show, Read, Eq, Ord, Functor, Foldable, Traversable, Typeable, Generic, Data)
instance Show1 f => Show1 (WrapF f) where
liftShowsPrec sp sl d (WrapF x) = showsUnaryWith (liftShowsPrec sp sl) "WrapF" d x
instance Eq1 f => Eq1 (WrapF f) where
liftEq eq (WrapF x) (WrapF y) = liftEq eq x y
instance Ord1 f => Ord1 (WrapF f) where
liftCompare c (WrapF x) (WrapF y) = liftCompare c x y
instance Tensor t i => Tensor (WrapHBF t) (WrapF i) where
type ListBy (WrapHBF t) = ListBy t
intro1 = WrapHBF . hright WrapF . intro1
intro2 = WrapHBF . hleft WrapF . intro2
elim1 = elim1 . hright unwrapF . unwrapHBF
elim2 = elim2 . hleft unwrapF . unwrapHBF
appendLB = appendLB . unwrapHBF
splitNE = WrapHBF . splitNE
splittingLB = splittingLB @t
. overHBifunctor (isoF WrapF unwrapF) (isoF WrapHBF unwrapHBF)
toListBy = toListBy . unwrapHBF
fromNE = fromNE @t
-- | Any @'ListBy' t f@ is a @'SemigroupIn' t@ and a @'MonoidIn' t i@, if we
-- have @'Tensor' t i@. This newtype wrapper witnesses that fact. We
-- require a newtype wrapper to avoid overlapping instances.
newtype WrapLB t f a = WrapLB { unwrapLB :: ListBy t f a }
instance Functor (ListBy t f) => Functor (WrapLB t f) where
fmap f (WrapLB x) = WrapLB (fmap f x)
-- | @since 0.3.0.0
instance Contravariant (ListBy t f) => Contravariant (WrapLB t f) where
contramap f (WrapLB x) = WrapLB (contramap f x)
-- | @since 0.3.0.0
instance Invariant (ListBy t f) => Invariant (WrapLB t f) where
invmap f g (WrapLB x) = WrapLB (invmap f g x)
instance (Tensor t i, FunctorBy t f, FunctorBy t (WrapLB t f)) => SemigroupIn (WrapHBF t) (WrapLB t f) where
biretract = WrapLB . appendLB . hbimap unwrapLB unwrapLB . unwrapHBF
binterpret f g = biretract . hbimap f g
instance (Tensor t i, FunctorBy t f, FunctorBy t (WrapLB t f)) => MonoidIn (WrapHBF t) (WrapF i) (WrapLB t f) where
pureT = WrapLB . nilLB @t . unwrapF