-- | Import this module qualified, like this:
--
-- > import qualified Rank2
--
-- This will bring into scope the standard classes 'Functor', 'Applicative', 'Foldable', and 'Traversable', but with a
-- @Rank2.@ prefix and a twist that their methods operate on a heterogenous collection. The same property is shared by
-- the two less standard classes 'Apply' and 'Distributive'.
{-# LANGUAGE InstanceSigs, KindSignatures, Rank2Types, ScopedTypeVariables, PolyKinds, DefaultSignatures #-}
module Rank2 (
-- * Rank 2 classes
Functor(..), Apply(..), Applicative(..),
Foldable(..), Traversable(..), Distributive(..), DistributiveTraversable(..), distributeJoin,
-- * Rank 2 data types
Compose(..), Empty(..), Only(..), Identity(..), Product(..), Arrow(..),
-- * Method synonyms and helper functions
ap, fmap, liftA3, liftA4, liftA5,
fmapTraverse, liftA2Traverse1, liftA2Traverse2, liftA2TraverseBoth,
cotraverse, cotraverseTraversable)
where
import qualified Control.Applicative as Rank1
import qualified Control.Monad as Rank1
import qualified Data.Foldable as Rank1
import qualified Data.Traversable as Rank1
import Data.Monoid (Monoid(..), (<>))
import Data.Functor.Compose (Compose(..))
import Prelude hiding (Foldable(..), Traversable(..), Functor(..), Applicative(..), (<$>), fst, snd)
-- | Equivalent of 'Functor' for rank 2 data types
class Functor g where
(<$>) :: (forall a. p a -> q a) -> g p -> g q
-- | Alphabetical synonym for '<$>'
fmap :: Functor g => (forall a. p a -> q a) -> g p -> g q
fmap f g = f <$> g
{-# INLINE fmap #-}
-- | Equivalent of 'Foldable' for rank 2 data types
class Foldable g where
foldMap :: Monoid m => (forall a. p a -> m) -> g p -> m
-- | Equivalent of 'Traversable' for rank 2 data types
class (Functor g, Foldable g) => Traversable g where
{-# MINIMAL traverse | sequence #-}
traverse :: Rank1.Applicative m => (forall a. p a -> m (q a)) -> g p -> m (g q)
sequence :: Rank1.Applicative m => g (Compose m p) -> m (g p)
traverse f = sequence . fmap (Compose . f)
sequence = traverse getCompose
-- | Wrapper for functions that map the argument constructor type
newtype Arrow p q a = Arrow{apply :: p a -> q a}
-- | Subclass of 'Functor' halfway to 'Applicative'
--
-- > (.) <$> u <*> v <*> w == u <*> (v <*> w)
class Functor g => Apply g where
{-# MINIMAL liftA2 | (<*>) #-}
-- | Equivalent of 'Rank1.<*>' for rank 2 data types
(<*>) :: g (Arrow p q) -> g p -> g q
-- | Equivalent of 'Rank1.liftA2' for rank 2 data types
liftA2 :: (forall a. p a -> q a -> r a) -> g p -> g q -> g r
-- | Equivalent of 'Rank1.liftA3' for rank 2 data types
liftA3 :: (forall a. p a -> q a -> r a -> s a) -> g p -> g q -> g r -> g s
(<*>) = liftA2 apply
liftA2 f g h = (Arrow . f) <$> g <*> h
liftA3 f g h i = liftA2 (\p q-> Arrow (f p q)) g h <*> i
liftA4 :: Apply g => (forall a. p a -> q a -> r a -> s a -> t a) -> g p -> g q -> g r -> g s -> g t
liftA4 f g h i j = liftA3 (\p q r-> Arrow (f p q r)) g h i <*> j
liftA5 :: Apply g => (forall a. p a -> q a -> r a -> s a -> t a -> u a) -> g p -> g q -> g r -> g s -> g t -> g u
liftA5 f g1 g2 g3 g4 g5 = liftA4 (\p q r s-> Arrow (f p q r s)) g1 g2 g3 g4 <*> g5
-- | Alphabetical synonym for '<*>'
ap :: Apply g => g (Arrow p q) -> g p -> g q
ap = (<*>)
-- | Equivalent of 'Rank1.Applicative' for rank 2 data types
class Apply g => Applicative g where
pure :: (forall a. f a) -> g f
-- | Equivalent of 'Rank1.Distributive' for rank 2 data types
class DistributiveTraversable g => Distributive g where
{-# MINIMAL distributeWith #-}
collect :: Rank1.Functor f1 => (a -> g f2) -> f1 a -> g (Compose f1 f2)
distribute :: Rank1.Functor f1 => f1 (g f2) -> g (Compose f1 f2)
distributeWith :: Rank1.Functor f1 => (forall x. f1 (f2 x) -> f x) -> f1 (g f2) -> g f
collect f = distribute . Rank1.fmap f
distribute = distributeWith Compose
-- | A weaker 'Distributive' that requires 'Rank1.Traversable' to use, not just a 'Rank1.Functor'.
class Functor g => DistributiveTraversable (g :: (k -> *) -> *) where
collectTraversable :: Rank1.Traversable f1 => (a -> g f2) -> f1 a -> g (Compose f1 f2)
distributeTraversable :: Rank1.Traversable f1 => f1 (g f2) -> g (Compose f1 f2)
distributeWithTraversable :: Rank1.Traversable f1 => (forall x. f1 (f2 x) -> f x) -> f1 (g f2) -> g f
collectTraversable f = distributeTraversable . Rank1.fmap f
distributeTraversable = distributeWithTraversable Compose
default distributeWithTraversable :: (Rank1.Traversable f1, Distributive g) =>
(forall x. f1 (f2 x) -> f x) -> f1 (g f2) -> g f
distributeWithTraversable = distributeWith
-- | A variant of 'distribute' convenient with 'Rank1.Monad' instances
distributeJoin :: (Distributive g, Rank1.Monad f) => f (g f) -> g f
distributeJoin = distributeWith Rank1.join
-- | Like 'fmap', but traverses over its argument
fmapTraverse :: (DistributiveTraversable f, Rank1.Traversable g) => (forall a. g (t a) -> u a) -> g (f t) -> f u
fmapTraverse f x = fmap (f . getCompose) (distributeTraversable x)
-- | Like 'liftA2', but traverses over its first argument
liftA2Traverse1 :: (Apply f, DistributiveTraversable f, Rank1.Traversable g) =>
(forall a. g (t a) -> u a -> v a) -> g (f t) -> f u -> f v
liftA2Traverse1 f x = liftA2 (f . getCompose) (distributeTraversable x)
-- | Like 'liftA2', but traverses over its second argument
liftA2Traverse2 :: (Apply f, DistributiveTraversable f, Rank1.Traversable g) =>
(forall a. t a -> g (u a) -> v a) -> f t -> g (f u) -> f v
liftA2Traverse2 f x y = liftA2 (\x' y' -> f x' (getCompose y')) x (distributeTraversable y)
-- | Like 'liftA2', but traverses over both its arguments
liftA2TraverseBoth :: (Apply f, DistributiveTraversable f, Rank1.Traversable g1, Rank1.Traversable g2) =>
(forall a. g1 (t a) -> g2 (u a) -> v a) -> g1 (f t) -> g2 (f u) -> f v
liftA2TraverseBoth f x y = liftA2 applyCompose (distributeTraversable x) (distributeTraversable y)
where applyCompose x' y' = f (getCompose x') (getCompose y')
-- | Equivalent of 'Rank1.cotraverse' for rank 2 data types
cotraverse :: (Distributive g, Rank1.Functor f) => (forall i. f (a i) -> b i) -> f (g a) -> g b
cotraverse f = (fmap (f . getCompose)) . distribute
-- | Equivalent of 'Rank1.cotraverse' for rank 2 data types using traversable
cotraverseTraversable :: (DistributiveTraversable g, Rank1.Traversable f) =>
(forall i. f (a i) -> b i) -> f (g a) -> g b
cotraverseTraversable f = (fmap (f . getCompose)) . distributeTraversable
-- | A rank-2 equivalent of '()', a zero-element tuple
data Empty f = Empty deriving (Eq, Ord, Show)
-- | A rank-2 tuple of only one element
newtype Only a f = Only {fromOnly :: f a} deriving (Eq, Ord, Show)
-- | Equivalent of 'Data.Functor.Identity' for rank 2 data types
newtype Identity g f = Identity {runIdentity :: g f} deriving (Eq, Ord, Show)
-- | Equivalent of 'Data.Functor.Product' for rank 2 data types
data Product g h f = Pair {fst :: g f, snd :: h f}
deriving (Eq, Ord, Show)
newtype Flip g a f = Flip (g (f a)) deriving (Eq, Ord, Show)
instance Monoid (g (f a)) => Monoid (Flip g a f) where
mempty = Flip mempty
Flip x `mappend` Flip y = Flip (x `mappend` y)
instance Rank1.Functor g => Rank2.Functor (Flip g a) where
f <$> Flip g = Flip (f Rank1.<$> g)
instance Rank1.Applicative g => Rank2.Apply (Flip g a) where
Flip g <*> Flip h = Flip (apply Rank1.<$> g Rank1.<*> h)
instance Rank1.Applicative g => Rank2.Applicative (Flip g a) where
pure f = Flip (Rank1.pure f)
instance Rank1.Foldable g => Rank2.Foldable (Flip g a) where
foldMap f (Flip g) = Rank1.foldMap f g
instance Rank1.Traversable g => Rank2.Traversable (Flip g a) where
traverse f (Flip g) = Flip Rank1.<$> Rank1.traverse f g
instance Functor Empty where
_ <$> _ = Empty
instance Functor (Only a) where
f <$> Only a = Only (f a)
instance Functor g => Functor (Identity g) where
f <$> Identity g = Identity (f <$> g)
instance (Functor g, Functor h) => Functor (Product g h) where
f <$> g = Pair (f <$> fst g) (f <$> snd g)
instance Foldable Empty where
foldMap _ _ = mempty
instance Foldable (Only x) where
foldMap f (Only x) = f x
instance Foldable g => Foldable (Identity g) where
foldMap f (Identity g) = foldMap f g
instance (Foldable g, Foldable h) => Foldable (Product g h) where
foldMap f ~(Pair g h) = foldMap f g <> foldMap f h
instance Traversable Empty where
traverse _ _ = Rank1.pure Empty
instance Traversable (Only x) where
traverse f (Only x) = Only Rank1.<$> f x
instance Traversable g => Traversable (Identity g) where
traverse f (Identity g) = Identity Rank1.<$> traverse f g
instance (Traversable g, Traversable h) => Traversable (Product g h) where
traverse f ~(Pair g h) = Rank1.liftA2 Pair (traverse f g) (traverse f h)
instance Apply Empty where
_ <*> _ = Empty
liftA2 _ _ _ = Empty
instance Apply (Only x) where
Only f <*> Only x = Only (apply f x)
liftA2 f (Only x) (Only y) = Only (f x y)
instance Apply g => Apply (Identity g) where
Identity g <*> Identity h = Identity (g <*> h)
liftA2 f (Identity g) (Identity h) = Identity (liftA2 f g h)
instance (Apply g, Apply h) => Apply (Product g h) where
gf <*> gx = Pair (fst gf <*> fst gx) (snd gf <*> snd gx)
liftA2 f ~(Pair g1 g2) ~(Pair h1 h2) = Pair (liftA2 f g1 h1) (liftA2 f g2 h2)
instance Applicative Empty where
pure = const Empty
instance Applicative (Only x) where
pure = Only
instance Applicative g => Applicative (Identity g) where
pure f = Identity (pure f)
instance (Applicative g, Applicative h) => Applicative (Product g h) where
pure f = Pair (pure f) (pure f)
instance DistributiveTraversable Empty
instance DistributiveTraversable (Only x)
instance DistributiveTraversable g => DistributiveTraversable (Identity g) where
distributeWithTraversable w f = Identity (distributeWithTraversable w $ Rank1.fmap runIdentity f)
instance (DistributiveTraversable g, DistributiveTraversable h) => DistributiveTraversable (Product g h) where
distributeWithTraversable w f = Pair (distributeWithTraversable w $ Rank1.fmap fst f)
(distributeWithTraversable w $ Rank1.fmap snd f)
instance Distributive Empty where
distributeWith _ _ = Empty
instance Distributive (Only x) where
distributeWith w f = Only (w $ Rank1.fmap fromOnly f)
instance Distributive g => Distributive (Identity g) where
distributeWith w f = Identity (distributeWith w $ Rank1.fmap runIdentity f)
instance (Distributive g, Distributive h) => Distributive (Product g h) where
distributeWith w f = Pair (distributeWith w $ Rank1.fmap fst f) (distributeWith w $ Rank1.fmap snd f)