{-# LANGUAGE DeriveDataTypeable #-} {-# LANGUAGE DeriveGeneric #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE Rank2Types #-} {-# LANGUAGE Safe #-} {-# LANGUAGE StandaloneDeriving #-} -------------------------------------------------------------------------------- -- | -- \"Applicative Effects in Free Monads\" -- -- Often times, the '(\<*\>)' operator can be more efficient than 'ap'. -- Conventional free monads don't provide any means of modeling this. -- The free monad can be modified to make use of an underlying applicative. -- But it does require some laws, or else the '(\<*\>)' = 'ap' law is broken. -- When interpreting this free monad with 'foldFree', -- the natural transformation must be an applicative homomorphism. -- An applicative homomorphism @hm :: (Applicative f, Applicative g) => f x -> g x@ -- will satisfy these laws. -- -- * @hm (pure a) = pure a@ -- * @hm (f \<*\> a) = hm f \<*\> hm a@ -- -- This is based on the \"Applicative Effects in Free Monads\" series of articles by Will Fancher -- -- * -- -- * -------------------------------------------------------------------------------- module Control.Monad.Free.Ap ( MonadFree(..) , Free(..) , retract , liftF , iter , iterA , iterM , hoistFree , foldFree , toFreeT , cutoff , unfold , unfoldM , _Pure, _Free ) where import Control.Applicative import Control.Arrow ((>>>)) import Control.Monad (liftM, MonadPlus(..), (>=>)) import Control.Monad.Fix import Control.Monad.Trans.Class import qualified Control.Monad.Trans.Free.Ap as FreeT import Control.Monad.Free.Class import Control.Monad.Reader.Class import Control.Monad.Writer.Class import Control.Monad.State.Class import Control.Monad.Error.Class import Control.Monad.Cont.Class import Data.Functor.Bind import Data.Functor.Classes import Data.Foldable import Data.Profunctor import Data.Traversable import Data.Semigroup.Foldable import Data.Semigroup.Traversable import Data.Data import GHC.Generics import Prelude hiding (foldr) -- $setup -- >>> import Control.Applicative (Const (..)) -- >>> import Data.Functor.Identity (Identity (..)) -- >>> import Data.Monoid (First (..)) -- >>> import Data.Tagged (Tagged (..)) -- >>> let preview l x = getFirst (getConst (l (Const . First . Just) x)) -- >>> let review l x = runIdentity (unTagged (l (Tagged (Identity x)))) -- | A free monad given an applicative data Free f a = Pure a | Free (f (Free f a)) deriving (Generic, Generic1) deriving instance ( Typeable f , Data a, Data (f (Free f a)) ) => Data (Free f a) instance Eq1 f => Eq1 (Free f) where liftEq eq = go where go (Pure a) (Pure b) = eq a b go (Free fa) (Free fb) = liftEq go fa fb go _ _ = False instance (Eq1 f, Eq a) => Eq (Free f a) where (==) = eq1 instance Ord1 f => Ord1 (Free f) where liftCompare cmp = go where go (Pure a) (Pure b) = cmp a b go (Pure _) (Free _) = LT go (Free _) (Pure _) = GT go (Free fa) (Free fb) = liftCompare go fa fb instance (Ord1 f, Ord a) => Ord (Free f a) where compare = compare1 instance Show1 f => Show1 (Free f) where liftShowsPrec sp sl = go where go d (Pure a) = showsUnaryWith sp "Pure" d a go d (Free fa) = showsUnaryWith (liftShowsPrec go (liftShowList sp sl)) "Free" d fa instance (Show1 f, Show a) => Show (Free f a) where showsPrec = showsPrec1 instance Read1 f => Read1 (Free f) where liftReadsPrec rp rl = go where go = readsData $ readsUnaryWith rp "Pure" Pure `mappend` readsUnaryWith (liftReadsPrec go (liftReadList rp rl)) "Free" Free instance (Read1 f, Read a) => Read (Free f a) where readsPrec = readsPrec1 instance Functor f => Functor (Free f) where fmap f = go where go (Pure a) = Pure (f a) go (Free fa) = Free (go <$> fa) {-# INLINE fmap #-} instance Apply f => Apply (Free f) where Pure a <.> Pure b = Pure (a b) Pure a <.> Free fb = Free $ fmap a <$> fb Free fa <.> Pure b = Free $ fmap ($ b) <$> fa Free fa <.> Free fb = Free $ fmap (<.>) fa <.> fb instance Applicative f => Applicative (Free f) where pure = Pure {-# INLINE pure #-} Pure a <*> Pure b = Pure $ a b Pure a <*> Free mb = Free $ fmap a <$> mb Free ma <*> Pure b = Free $ fmap ($ b) <$> ma Free ma <*> Free mb = Free $ fmap (<*>) ma <*> mb instance Apply f => Bind (Free f) where Pure a >>- f = f a Free m >>- f = Free ((>>- f) <$> m) instance Applicative f => Monad (Free f) where return = pure {-# INLINE return #-} Pure a >>= f = f a Free m >>= f = Free ((>>= f) <$> m) instance Applicative f => MonadFix (Free f) where mfix f = a where a = f (impure a); impure (Pure x) = x; impure (Free _) = error "mfix (Free f): Free" -- | This violates the Alternative laws, handle with care. instance Alternative v => Alternative (Free v) where empty = Free empty {-# INLINE empty #-} a <|> b = Free (pure a <|> pure b) {-# INLINE (<|>) #-} -- | This violates the MonadPlus laws, handle with care. instance MonadPlus v => MonadPlus (Free v) where mzero = Free mzero {-# INLINE mzero #-} a `mplus` b = Free (return a `mplus` return b) {-# INLINE mplus #-} -- | This is not a true monad transformer. It is only a monad transformer \"up to 'retract'\". instance MonadTrans Free where lift = Free . liftM Pure {-# INLINE lift #-} instance Foldable f => Foldable (Free f) where foldMap f = go where go (Pure a) = f a go (Free fa) = foldMap go fa {-# INLINE foldMap #-} foldr f = go where go r free = case free of Pure a -> f a r Free fa -> foldr (flip go) r fa {-# INLINE foldr #-} foldl' f = go where go r free = case free of Pure a -> f r a Free fa -> foldl' go r fa {-# INLINE foldl' #-} instance Foldable1 f => Foldable1 (Free f) where foldMap1 f = go where go (Pure a) = f a go (Free fa) = foldMap1 go fa {-# INLINE foldMap1 #-} instance Traversable f => Traversable (Free f) where traverse f = go where go (Pure a) = Pure <$> f a go (Free fa) = Free <$> traverse go fa {-# INLINE traverse #-} instance Traversable1 f => Traversable1 (Free f) where traverse1 f = go where go (Pure a) = Pure <$> f a go (Free fa) = Free <$> traverse1 go fa {-# INLINE traverse1 #-} instance MonadWriter e m => MonadWriter e (Free m) where tell = lift . tell {-# INLINE tell #-} listen = lift . listen . retract {-# INLINE listen #-} pass = lift . pass . retract {-# INLINE pass #-} instance MonadReader e m => MonadReader e (Free m) where ask = lift ask {-# INLINE ask #-} local f = lift . local f . retract {-# INLINE local #-} instance MonadState s m => MonadState s (Free m) where get = lift get {-# INLINE get #-} put s = lift (put s) {-# INLINE put #-} instance MonadError e m => MonadError e (Free m) where throwError = lift . throwError {-# INLINE throwError #-} catchError as f = lift (catchError (retract as) (retract . f)) {-# INLINE catchError #-} instance MonadCont m => MonadCont (Free m) where callCC f = lift (callCC (retract . f . liftM lift)) {-# INLINE callCC #-} instance Applicative f => MonadFree f (Free f) where wrap = Free {-# INLINE wrap #-} -- | -- 'retract' is the left inverse of 'lift' and 'liftF' -- -- @ -- 'retract' . 'lift' = 'id' -- 'retract' . 'liftF' = 'id' -- @ retract :: Monad f => Free f a -> f a retract = foldFree id -- | Given an applicative homomorphism from @f@ to 'Identity', tear down a 'Free' 'Monad' using iteration. iter :: Applicative f => (f a -> a) -> Free f a -> a iter _ (Pure a) = a iter phi (Free m) = phi (iter phi <$> m) -- | Like 'iter' for applicative values. iterA :: (Applicative p, Applicative f) => (f (p a) -> p a) -> Free f a -> p a iterA _ (Pure x) = pure x iterA phi (Free f) = phi (iterA phi <$> f) -- | Like 'iter' for monadic values. iterM :: (Monad m, Applicative f) => (f (m a) -> m a) -> Free f a -> m a iterM _ (Pure x) = return x iterM phi (Free f) = phi (iterM phi <$> f) -- | Lift an applicative homomorphism from @f@ to @g@ into a monad homomorphism from @'Free' f@ to @'Free' g@. hoistFree :: (Applicative f, Applicative g) => (forall a. f a -> g a) -> Free f b -> Free g b hoistFree f = foldFree (liftF . f) -- | Given an applicative homomorphism, you get a monad homomorphism. foldFree :: (Applicative f, Monad m) => (forall x . f x -> m x) -> Free f a -> m a foldFree _ (Pure a) = return a foldFree f (Free as) = f as >>= foldFree f -- | Convert a 'Free' monad from "Control.Monad.Free.Ap" to a 'FreeT.FreeT' monad -- from "Control.Monad.Trans.Free.Ap". -- WARNING: This assumes that 'liftF' is an applicative homomorphism. toFreeT :: (Applicative f, Monad m) => Free f a -> FreeT.FreeT f m a toFreeT = foldFree liftF -- | Cuts off a tree of computations at a given depth. -- If the depth is 0 or less, no computation nor -- monadic effects will take place. -- -- Some examples (n ≥ 0): -- -- prop> cutoff 0 _ == return Nothing -- prop> cutoff (n+1) . return == return . Just -- prop> cutoff (n+1) . lift == lift . liftM Just -- prop> cutoff (n+1) . wrap == wrap . fmap (cutoff n) -- -- Calling 'retract . cutoff n' is always terminating, provided each of the -- steps in the iteration is terminating. cutoff :: (Applicative f) => Integer -> Free f a -> Free f (Maybe a) cutoff n _ | n <= 0 = return Nothing cutoff n (Free f) = Free $ fmap (cutoff (n - 1)) f cutoff _ m = Just <$> m -- | Unfold a free monad from a seed. unfold :: Applicative f => (b -> Either a (f b)) -> b -> Free f a unfold f = f >>> either Pure (Free . fmap (unfold f)) -- | Unfold a free monad from a seed, monadically. unfoldM :: (Applicative f, Traversable f, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a) unfoldM f = f >=> either (pure . pure) (fmap Free . traverse (unfoldM f)) -- | This is @Prism' (Free f a) a@ in disguise -- -- >>> preview _Pure (Pure 3) -- Just 3 -- -- >>> review _Pure 3 :: Free Maybe Int -- Pure 3 _Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a)) _Pure = dimap impure (either pure (fmap Pure)) . right' where impure (Pure x) = Right x impure x = Left x {-# INLINE impure #-} {-# INLINE _Pure #-} -- | This is @Prism' (Free f a) (f (Free f a))@ in disguise -- -- >>> preview _Free (review _Free (Just (Pure 3))) -- Just (Just (Pure 3)) -- -- >>> review _Free (Just (Pure 3)) -- Free (Just (Pure 3)) _Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a)) _Free = dimap unfree (either pure (fmap Free)) . right' where unfree (Free x) = Right x unfree x = Left x {-# INLINE unfree #-} {-# INLINE _Free #-}