{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
-- |
-- Module : Data.HFunctor.Final
-- Copyright : (c) Justin Le 2019
-- License : BSD3
--
-- Maintainer : justin@jle.im
-- Stability : experimental
-- Portability : non-portable
--
-- Provides 'Final', which can be considered the "free 'Interpret' over
-- a constraint": generate a handy 'Interpret' instance for any constraint
-- @c@.
module Data.HFunctor.Final (
Final(..)
, fromFinal, toFinal
, FreeOf(..), finalizing
, hoistFinalC
, liftFinal0
, liftFinal1
, liftFinal2
) where
import Control.Applicative
import Control.Applicative.Free
import Control.Applicative.Lift
import Control.Applicative.ListF
import Control.Monad
import Control.Monad.Freer.Church hiding (toFree)
import Control.Monad.Reader
import Control.Monad.Trans.Identity
import Control.Natural
import Control.Natural.IsoF
import Data.Constraint.Trivial
import Data.Functor.Apply.Free
import Data.Functor.Bind
import Data.Functor.Coyoneda
import Data.Functor.Plus
import Data.HFunctor
import Data.HFunctor.Interpret
import Data.Pointed
import qualified Control.Alternative.Free as Alt
import qualified Control.Applicative.Free.Fast as FAF
-- | A simple way to inject/reject into any eventual typeclass.
--
-- In a way, this is the "ultimate" multi-purpose 'Interpret' instance.
-- You can use this to inject an @f@ into a free structure of any
-- typeclass. If you want @f@ to have a 'Monad' instance, for example,
-- just use
--
-- @
-- 'inject' :: f a -> 'Final' 'Monad' f a
-- @
--
-- When you want to eventually interpret out the data, use:
--
-- @
-- 'interpret' :: (f '~>' g) -> 'Final' c f a -> g a
-- @
--
-- Essentially, @'Final' c@ is the "free c". @'Final' 'Monad'@ is the free
-- 'Monad', etc.
--
-- 'Final' can theoretically replace 'Ap', 'Ap1', 'ListF', 'NonEmptyF',
-- 'MaybeF', 'Free', 'Data.Functor.Identity.Identity', 'Coyoneda', and
-- other instances of 'FreeOf', if you don't care about being able to
-- pattern match on explicit structure.
--
-- However, it cannot replace 'Interpret' instances that are not free
-- structures, like 'Control.Applicative.Step.Step',
-- 'Control.Applicative.Step.Steps',
-- 'Control.Applicative.Backwards.Backwards', etc.
--
-- Note that this doesn't have instances for /all/ the typeclasses you
-- could lift things into; you probably have to define your own if you want
-- to use @'Final' c@ as an /instance/ of @c@ (using 'liftFinal0',
-- 'liftFinal1', 'liftFinal2' for help).
newtype Final c f a = Final
{ runFinal :: forall g. c g => (forall x. f x -> g x) -> g a }
-- | Lift an action into a 'Final'.
liftFinal0
:: (forall g. c g => g a)
-> Final c f a
liftFinal0 x = Final $ \_ -> x
-- | Map the action in a 'Final'.
liftFinal1
:: (forall g. c g => g a -> g b)
-> Final c f a
-> Final c f b
liftFinal1 f x = Final $ \r -> f (runFinal x r)
-- | Merge two 'Final' actions.
liftFinal2
:: (forall g. c g => g a -> g b -> g d)
-> Final c f a
-> Final c f b
-> Final c f d
liftFinal2 f x y = Final $ \r -> f (runFinal x r) (runFinal y r)
instance Functor (Final Functor f) where
fmap f = liftFinal1 (fmap f)
instance Functor (Final Apply f) where
fmap f = liftFinal1 (fmap f)
instance Apply (Final Apply f) where
(<.>) = liftFinal2 (<.>)
liftF2 f = liftFinal2 (liftF2 f)
instance Functor (Final Bind f) where
fmap f = liftFinal1 (fmap f)
instance Apply (Final Bind f) where
(<.>) = liftFinal2 (<.>)
liftF2 f = liftFinal2 (liftF2 f)
instance Bind (Final Bind f) where
x >>- f = Final $ \r -> runFinal x r >>- \y -> runFinal (f y) r
instance Functor (Final Applicative f) where
fmap f = liftFinal1 (fmap f)
instance Apply (Final Applicative f) where
(<.>) = liftFinal2 (<*>)
liftF2 f = liftFinal2 (liftA2 f)
instance Applicative (Final Applicative f) where
pure x = liftFinal0 (pure x)
(<*>) = liftFinal2 (<*>)
liftA2 f = liftFinal2 (liftA2 f)
instance Functor (Final Alternative f) where
fmap f = liftFinal1 (fmap f)
instance Apply (Final Alternative f) where
(<.>) = liftFinal2 (<*>)
liftF2 f = liftFinal2 (liftA2 f)
instance Applicative (Final Alternative f) where
pure x = liftFinal0 (pure x)
(<*>) = liftFinal2 (<*>)
liftA2 f = liftFinal2 (liftA2 f)
instance Alternative (Final Alternative f) where
empty = liftFinal0 empty
(<|>) = liftFinal2 (<|>)
instance Functor (Final Monad f) where
fmap f = liftFinal1 (fmap f)
instance Apply (Final Monad f) where
(<.>) = liftFinal2 (<*>)
liftF2 f = liftFinal2 (liftA2 f)
instance Applicative (Final Monad f) where
pure x = liftFinal0 (pure x)
(<*>) = liftFinal2 (<*>)
liftA2 f = liftFinal2 (liftA2 f)
instance Monad (Final Monad f) where
return x = liftFinal0 (return x)
x >>= f = Final $ \r -> do
y <- runFinal x r
runFinal (f y) r
instance Functor (Final MonadPlus f) where
fmap f = liftFinal1 (fmap f)
instance Applicative (Final MonadPlus f) where
pure x = liftFinal0 (pure x)
(<*>) = liftFinal2 (<*>)
liftA2 f = liftFinal2 (liftA2 f)
instance Monad (Final MonadPlus f) where
return x = liftFinal0 (return x)
x >>= f = Final $ \r -> do
y <- runFinal x r
runFinal (f y) r
instance Alternative (Final MonadPlus f) where
empty = liftFinal0 empty
(<|>) = liftFinal2 (<|>)
instance MonadPlus (Final MonadPlus f) where
mzero = liftFinal0 mzero
mplus = liftFinal2 mplus
instance Pointed (Final Pointed f) where
point x = liftFinal0 (point x)
instance Functor (Final (MonadReader r) f) where
fmap f = liftFinal1 (fmap f)
instance Applicative (Final (MonadReader r) f) where
pure x = liftFinal0 (pure x)
(<*>) = liftFinal2 (<*>)
liftA2 f = liftFinal2 (liftA2 f)
instance Apply (Final (MonadReader r) f) where
(<.>) = liftFinal2 (<*>)
liftF2 f = liftFinal2 (liftA2 f)
instance Monad (Final (MonadReader r) f) where
return x = liftFinal0 (return x)
x >>= f = Final $ \r -> do
y <- runFinal x r
runFinal (f y) r
instance MonadReader r (Final (MonadReader r) f) where
ask = liftFinal0 ask
local f = liftFinal1 (local f)
instance Functor (Final Alt f) where
fmap f = liftFinal1 (fmap f)
instance Alt (Final Alt f) where
(<!>) = liftFinal2 (<!>)
instance Functor (Final Plus f) where
fmap f = liftFinal1 (fmap f)
instance Alt (Final Plus f) where
(<!>) = liftFinal2 (<!>)
instance Plus (Final Plus f) where
zero = liftFinal0 zero
-- | Re-interpret the context under a 'Final'.
hoistFinalC
:: (forall g x. (c g => g x) -> (d g => g x))
-> Final c f a
-> Final d f a
hoistFinalC f (Final x) = Final $ \r -> f (x (\y -> f (r y)))
instance HFunctor (Final c) where
hmap f x = Final $ \r -> runFinal x (r . f)
instance Inject (Final c) where
inject x = Final ($ x)
instance Interpret (Final c) where
type C (Final c) = c
retract x = runFinal x id
interpret f x = runFinal x f
-- | "Finalize" an 'Interpret' instance.
--
-- @
-- toFinal :: 'Coyoneda' f '~>' 'Final' 'Functor' f
-- toFinal :: 'Ap' f '~>' 'Final' 'Applicative' f
-- toFinal :: 'Alt' f '~>' 'Final' 'Alternative' f
-- toFinal :: 'Free' f '~>' 'Final' 'Monad' f
-- toFinal :: 'Lift' f '~>' 'Final' 'Pointed' f
-- toFinal :: 'ListF' f '~>' 'Final' 'Plus' f
-- @
--
-- Note that the instance of @c@ for @'Final' c@ must be defined.
--
-- This operation can potentially /forget/ structure in @t@. For example,
-- we have:
--
-- @
-- 'toFinal' :: 'Control.Applicative.Step.Steps' f ~> 'Final' 'Alt' f
-- @
--
-- In this process, we lose the "positional" structure of
-- 'Control.Applicative.Step.Steps'.
--
-- In the case where 'toFinal' doesn't lose any information, this will form
-- an isomorphism with 'fromFinal', and @t@ is known as the "Free @c@".
-- For such a situation, @t@ will have a 'FreeOf' instance.
toFinal :: (Interpret t, C t (Final c f)) => t f ~> Final c f
toFinal = interpret inject
-- | "Concretize" a 'Final'.
-- @
-- fromFinal :: 'Final' 'Functor' f '~>' 'Coyoneda' f
-- fromFinal :: 'Final' 'Applicative' f '~>' 'Ap' f
-- fromFinal :: 'Final' 'Alternative' f '~>' 'Alt' f
-- fromFinal :: 'Final' 'Monad' f '~>' 'Free' f
-- fromFinal :: 'Final' 'Pointed' f '~>' 'Lift' f
-- fromFinal :: 'Final' 'Plus' f '~>' 'ListF' f
-- @
--
-- This can be useful because 'Final' doesn't have a concrete structure
-- that you can pattern match on and inspect, but @t@ might.
--
-- In the case that this forms an isomorphism with 'toFinal', the @t@ will
-- have an instance of 'FreeOf'.
fromFinal :: (Interpret t, c (t f)) => Final c f ~> t f
fromFinal = interpret inject
-- | A typeclass associating a free structure with the typeclass it is free
-- on.
--
-- This essentially lists instances of 'Interpret' where a "trip" through
-- 'Final' will leave it unchanged.
--
-- @
-- 'fromFree' . 'toFree' == id
-- 'toFree' . 'fromFree' == id
-- @
--
-- This can be useful because 'Final' doesn't have a concrete structure
-- that you can pattern match on and inspect, but @t@ might. This lets you
-- work on a concrete structure if you desire.
class Interpret t => FreeOf c t | t -> c where
fromFree :: t f ~> Final c f
toFree :: Functor f => Final c f ~> t f
default fromFree :: C t (Final c f) => t f ~> Final c f
fromFree = toFinal
default toFree :: c (t f) => Final c f ~> t f
toFree = fromFinal
-- | The isomorphism between a free structure and its encoding as 'Final'.
finalizing :: (FreeOf c t, Functor f) => t f <~> Final c f
finalizing = isoF fromFree toFree
instance FreeOf Functor Coyoneda
instance FreeOf Applicative Ap
instance FreeOf Apply Ap1
instance FreeOf Applicative FAF.Ap
instance FreeOf Alternative Alt.Alt
instance FreeOf Monad Free
instance FreeOf Bind Free1
instance FreeOf Pointed Lift
instance FreeOf Pointed MaybeApply
instance FreeOf Alt NonEmptyF
instance FreeOf Plus ListF
instance FreeOf Unconstrained IdentityT