{-# LANGUAGE GADTs #-} {-# LANGUAGE GeneralizedNewtypeDeriving #-} module Control.Algebra.Free ( -- Higher free algebra class FreeAlgebra1 (..) -- ** Type level witnesses , Proof (..) , proof -- ** Higher algebra type , AlgebraType0 , AlgebraType -- * Combinators , wrapFree , foldFree1 , unFoldNatFree , hoistFree1 , hoistFreeH , joinFree1 , bindFree1 , assocFree1 , iterFree1 , cataFree1 -- * Day convolution , DayF (..) , dayToAp , apToDay -- * Various classes (higher algebra types) , MonadList (..) , MonadMaybe (..) ) where import Control.Applicative (Alternative) import Control.Applicative.Free (Ap) import qualified Control.Applicative.Free as Ap import qualified Control.Applicative.Free.Fast as Fast import qualified Control.Applicative.Free.Final as Final import Control.Alternative.Free (Alt (..)) import qualified Control.Alternative.Free as Alt import Control.Monad (foldM, join) import Control.Monad.Except (ExceptT (..), MonadError (..)) import Control.Monad.Free (Free) import qualified Control.Monad.Free as Free import qualified Control.Monad.Free.Church as Church import Control.Monad.List (ListT (..)) import Control.Monad.Reader (MonadReader (..), ReaderT (..)) import Control.Monad.RWS.Class (MonadRWS) import Control.Monad.RWS.Lazy as L (RWST (..)) import Control.Monad.RWS.Strict as S (RWST (..)) import Control.Monad.State.Class (MonadState (..)) import qualified Control.Monad.State.Lazy as L (StateT (..)) import qualified Control.Monad.State.Strict as S (StateT (..)) import Control.Monad.Trans.Class (MonadTrans (..)) import Control.Monad.Trans.Maybe (MaybeT (..)) import Control.Monad.Writer.Class (MonadWriter (..)) import qualified Control.Monad.Writer.Lazy as L (WriterT (..)) import qualified Control.Monad.Writer.Strict as S (WriterT (..)) import Data.Constraint (Dict (..)) import Data.Kind (Type) import Data.Fix (Fix, cataM) import Data.Functor.Coyoneda (Coyoneda (..), liftCoyoneda) import Data.Functor.Day (Day (..)) import qualified Data.Functor.Day as Day import Data.Functor.Identity (Identity (..)) import Data.Algebra.Free (AlgebraType, AlgebraType0, Proof (..), proof) -- | -- Higher kinded version of @'FreeAlgebra'@. Instances includes free functors, -- free applicative functors, free monads, state monads etc. -- -- A lawful instance should guarantee that @'foldNatFree'@ is an isomorphism -- with inverses @'unFoldNatFree'@. -- -- This guaranties that @m@ is a left adjoint functor from the category of -- types of kind @Type -> Type@ which satisfy @'AlgebraType0' m@ constraint, to the -- category of types of kind @Type -> Type@ which satisfy the @'AlgebraType' m@ -- constraint. This functor is left adjoin to the forgetful functor (which is -- well defined if the laws on @'AlgebraType0'@ family are satisfied. This in -- turn guarantees that @m@ composed with this forgetful functor is a monad. -- In result we get monadic operations: -- -- * @return = 'liftFree'@ -- * @(>>=) = 'bindFree1'@ -- * @join = 'joinFree1'@ -- -- For @m@ such that @'AlgebraType0'@ subsumes @'Monad'@ this class implies: -- -- * @MFunctor@ via @hoist = hoistFree1@ -- * @MMonad@ via @embed = flip bindFree1@ -- * @MonadTrans@ via @lift = liftFree@ class FreeAlgebra1 (m :: (k -> Type) -> k -> Type) where -- | Natural transformation that embeds generators into @m@. liftFree :: AlgebraType0 m f => f a -> m f a -- | The freeness property. foldNatFree :: forall d f a . ( AlgebraType m d , AlgebraType0 m f ) => (forall x. f x -> d x) -- ^ a natural transformation which embeds generators of @m@ into @d@ -> (m f a -> d a) -- ^ a morphism from @m f@ to @d@ -- | -- A proof that @'AlgebraType' m (m f)@ holds for all @AlgebraType0 f => f@. -- Together with @'hoistFree1'@ this proves that @FreeAlgebra m => m@ is -- a functor from the full subcategory of types of kind @Type -> Type@ -- which satisfy @'AlgebraType0' m f@ to ones that satisfy @'AlgebraType' -- m f@. codom1 :: forall f. AlgebraType0 m f => Proof (AlgebraType m (m f)) (m f) -- | -- A proof that the forgetful functor from the full subcategory of types of -- kind @Type -> Type@ satisfying @'AlgebraType' m f@ constraint to types -- satisfying @'AlgebraType0' m f@ is well defined. forget1 :: forall f. AlgebraType m f => Proof (AlgebraType0 m f) (m f) -- | -- Anything that carries @'FreeAlgebra1'@ constraint is also an instance of -- @'Control.Monad.Free.Class.MonadFree'@, but not vice versa. You can use -- @'wrap'@ to define a @'Control.Monad.Free.Class.MonadFree'@ instance. -- @'ContT'@ is an example of a monad which does have an @'FreeAlgebra1'@ -- instance, but has an @'MonadFree'@ instance. -- -- The @'Monad'@ constrain will be satisfied for many monads through the -- @'AlgebraType m'@ constraint. wrapFree :: forall (m :: (Type -> Type) -> Type -> Type) (f :: Type -> Type) a . ( FreeAlgebra1 m , AlgebraType0 m f , Monad (m f) ) => f (m f a) -> m f a wrapFree = join . liftFree {-# INLINE wrapFree #-} -- | -- @'FreeAlgebra1' m@ implies that @m f@ is a foldable. -- -- @ -- 'foldFree1' . 'liftFree' == 'id' :: f a -> f a -- @ -- -- @foldFree1@ is the -- [unit](https://ncatlab.org/nlab/show/unit+of+an+adjunction) of the -- adjunction imposed by @FreeAlgebra1@ constraint. -- -- It can be specialized to: -- -- * @'Data.Functor.Coyoneda.lowerCoyoneda' :: 'Functor' f => 'Coyoneda' f a -> f a@ -- * @'Control.Applicative.Free.retractAp' :: 'Applicative' f => 'Ap' f a -> f a@ -- * @'Control.Monad.Free.foldFree' :: 'Monad' m => (forall x. f x -> m x) -> 'Free' f a -> m a@ foldFree1 :: forall m f a . ( FreeAlgebra1 m , AlgebraType m f ) => m f a -> f a foldFree1 = case forget1 :: Proof (AlgebraType0 m f) (m f) of Proof Dict -> foldNatFree id {-# INLINE foldFree1 #-} -- | -- @'unFoldNatFree'@ is an inverse of @'foldNatFree'@ -- -- prop> unFoldNatFree id = ruturnFree1 -- -- Note that @'unFoldNatFree' id@ is the -- [unit](https://ncatlab.org/nlab/show/unit+of+an+adjunction) of the -- adjunction imposed by the @'FreeAlgebra1'@ constraint. unFoldNatFree :: ( FreeAlgebra1 m , AlgebraType0 m f ) => (forall x . m f x -> d x) -> f a -> d a unFoldNatFree nat = nat . liftFree -- | -- This is a functor instance for @m@ when considered as an endofuctor of some -- subcategory of @Type -> Type@ (e.g. endofunctors of _Hask_). -- -- It can be specialized to: -- -- * @'Control.Applicative.Free.hoistAp' :: (forall a. f a -> g a) -> 'Ap' f b -> 'Ap' g b @ -- * @'Control.Monad.Free.hoistFree' :: 'Functor' g => (forall a. f a -> g a) -> 'Free' f b -> 'Free' g b@ -- * @Control.Monad.Morph.hoist@ for @'FreeAlgebra1' m => m@ such that -- @'AlgebraType0' m@ subsumes @Monad m@, e.g. -- @'Control.Monad.State.Lazy.StateT'@, @'Control.Monad.Writer.Lazy.WriterT'@ -- or @'Control.Monad.Reader.ReaderT'@. hoistFree1 :: forall m f g a . ( FreeAlgebra1 m , AlgebraType0 m g , AlgebraType0 m f ) => (forall x. f x -> g x) -- ^ a natural transformation @f ~> g@ -> m f a -> m g a hoistFree1 nat = case codom1 :: Proof (AlgebraType m (m g)) (m g) of Proof Dict -> foldNatFree (liftFree . nat) {-# INLINE hoistFree1 #-} -- | -- @ -- 'hoistFreeH' . 'hoistFreeH' = 'hoistFreeH' -- @ -- -- and when @'FreeAlgebra1' m ~ 'FreeAlgebra1' n@: -- -- @ -- 'hoistFreeH' = 'id' -- @ hoistFreeH :: forall m n f a . ( FreeAlgebra1 m , FreeAlgebra1 n , AlgebraType0 m f , AlgebraType0 n f , AlgebraType m (n f) ) => m f a -> n f a hoistFreeH = foldNatFree liftFree {-# INLINE hoistFreeH #-} -- | -- @'joinFree1'@ makes @m@ a monad in some subcatgory of types of kind @Type -> Type@ -- (usually the endo-functor category of @Hask@). It is just a specialization -- of @'foldFree1'@. joinFree1 :: forall m f a . ( FreeAlgebra1 m , AlgebraType0 m f ) => m (m f) a -> m f a joinFree1 = case codom1 :: Proof (AlgebraType m (m f)) (m f) of Proof Dict -> case forget1 :: Proof (AlgebraType0 m (m f)) (m (m f)) of Proof Dict -> foldFree1 {-# INLINE joinFree1 #-} -- | -- Bind operator for the @'joinFree1'@ monad, this is just @'foldNatFree'@ in -- disguise. -- -- For @'Control.Monad.State.Lazy.StateT'@, -- @'Control.Monad.Writer.Lazy.WriterT'@ or -- @'Control.Monad.Reader.Lazy.ReaderT'@ (or any @'FreeAlgebra1' m => m@ such -- that @'AlgebraType0' m@ subsumes @'Monad' m@), this is the @>>=@ version of -- @Control.Monad.Morph.embed@. bindFree1 :: forall m f g a . ( FreeAlgebra1 m , AlgebraType0 m g , AlgebraType0 m f ) => m f a -> (forall x . f x -> m g x) -- ^ natural transformation @f ~> m g@ -> m g a bindFree1 mfa nat = case codom1 :: Proof (AlgebraType m (m g)) (m g) of Proof Dict -> foldNatFree nat mfa {-# INLINE bindFree1 #-} assocFree1 :: forall m f a . ( FreeAlgebra1 m , AlgebraType m f , Functor (m (m f)) ) => m f (m f a) -> m (m f) (f a) assocFree1 = case forget1 :: Proof (AlgebraType0 m f) (m f) of Proof Dict -> case codom1 :: Proof (AlgebraType m (m f)) (m f) of Proof Dict -> case forget1 :: Proof (AlgebraType0 m (m f)) (m (m f)) of Proof Dict -> case codom1 :: Proof (AlgebraType m (m (m f))) (m (m f)) of Proof Dict -> fmap foldFree1 <$> foldNatFree (hoistFree1 liftFree . liftFree) {-# INLINE assocFree1 #-} -- | -- @'Fix' (m f)@ is the initial /algebra/ of type @'AlgebraType' m@ and -- @'AlgebraType0' f@. cataFree1 :: forall m f a . ( FreeAlgebra1 m , AlgebraType m f , Monad f , Traversable (m f) ) => Fix (m f) -> f a cataFree1 = cataM foldFree1 -- | -- Specialization of @'foldNatFree' \@_ \@'Identity'@; it will further specialize to: -- -- * @\\_ -> 'runIdentity' . 'Data.Functor.Coyoneda.lowerCoyoneda'@ -- * @'Control.Applicative.Free.iterAp' :: 'Functor' g => (g a -> a) -> 'Ap' g a -> a@ -- * @'Control.Monad.Free.iter' :: 'Functor' f => (f a -> a) -> 'Free' f a -> a@ iterFree1 :: forall m f a . ( FreeAlgebra1 m , AlgebraType0 m f , AlgebraType m Identity ) => (forall x . f x -> x) -> m f a -> a iterFree1 f = runIdentity . foldNatFree (Identity . f) {-# INLINE iterFree1 #-} -- Instances -- | -- Algebras of the same type as @'Coyoneda'@ are all functors. type instance AlgebraType0 Coyoneda g = () type instance AlgebraType Coyoneda g = Functor g instance FreeAlgebra1 Coyoneda where liftFree = liftCoyoneda foldNatFree nat (Coyoneda ba fx) = ba <$> nat fx codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'Ap'@ are the applicative functors. type instance AlgebraType0 Ap g = Functor g type instance AlgebraType Ap g = Applicative g -- | -- @'Ap'@ is a free in the class of applicative functors, over any functor -- (@'Ap' f@ is applicative whenever @f@ is a functor) instance FreeAlgebra1 Ap where liftFree = Ap.liftAp foldNatFree = Ap.runAp codom1 = proof forget1 = proof type instance AlgebraType0 Fast.Ap g = Functor g type instance AlgebraType Fast.Ap g = Applicative g instance FreeAlgebra1 Fast.Ap where liftFree = Fast.liftAp foldNatFree = Fast.runAp codom1 = proof forget1 = proof type instance AlgebraType0 Final.Ap g = Functor g type instance AlgebraType Final.Ap g = Applicative g instance FreeAlgebra1 Final.Ap where liftFree = Final.liftAp foldNatFree = Final.runAp codom1 = proof forget1 = proof -- | -- @'Day' f f@ newtype wrapper. It is isomorphic with @'Ap' f@ for applicative -- functors @f@ via @'dayToAp'@ (and @'apToDay'@). newtype DayF f a = DayF { runDayF :: Day f f a} deriving (Functor, Applicative) dayToAp :: Applicative f => Day f f a -> Ap f a dayToAp = hoistFreeH . DayF apToDay :: Applicative f => Ap f a -> Day f f a apToDay = runDayF . hoistFreeH -- | -- Algebras of the same type as @'DayF'@ are all the applicative functors. type instance AlgebraType0 DayF g = Applicative g type instance AlgebraType DayF g = Applicative g -- | -- @'DayF'@, as @'Ap'@ is a free applicative functor, but over applicative functors -- (@'DayF' f@ is applicative if @f@ is an applicative functor). instance FreeAlgebra1 DayF where liftFree fa = DayF $ Day fa fa const foldNatFree nat (DayF day) = Day.dap . Day.trans2 nat . Day.trans1 nat $ day codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'Free'@ monad is the class of all monads. type instance AlgebraType0 Free f = Functor f type instance AlgebraType Free m = Monad m -- | -- @'Free'@ monad is free in the class of monad over the class of functors. instance FreeAlgebra1 Free where liftFree = Free.liftF foldNatFree = Free.foldFree codom1 = proof forget1 = proof type instance AlgebraType0 Church.F f = Functor f type instance AlgebraType Church.F m = Monad m instance FreeAlgebra1 Church.F where liftFree = Church.liftF foldNatFree = Church.foldF codom1 = proof forget1 = proof type instance AlgebraType0 Alt f = Functor f type instance AlgebraType Alt m = Alternative m instance FreeAlgebra1 Alt where liftFree = Alt.liftAlt foldNatFree = Alt.runAlt codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'L.StateT'@ monad is the class of all state -- monads. type instance AlgebraType0 (L.StateT s) m = Monad m type instance AlgebraType (L.StateT s) m = ( MonadState s m ) -- | -- Lazy @'L.StateT'@ monad transformer is a free algebra in the class of monads -- which satisfy the @'MonadState'@ constraint. Note that this instance -- captures that @'L.StateT' s@ is a monad transformer: -- -- @ -- 'liftFree' = 'lift' -- @ -- -- This is also true for all the other monad transformers. instance FreeAlgebra1 (L.StateT s) where liftFree = lift foldNatFree nat ma = do (a, s) <- get >>= nat . L.runStateT ma put s return a codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'S.StateT'@ monad is the class of all state -- monads. type instance AlgebraType0 (S.StateT s) m = Monad m type instance AlgebraType (S.StateT s) m = ( MonadState s m ) -- | -- Strict @'S.StateT'@ monad transformer is also a free algebra, thus @'hoistFreeH'@ -- is an isomorphism between the strict and lazy versions. instance FreeAlgebra1 (S.StateT s) where liftFree :: Monad m => m a -> S.StateT s m a liftFree = lift foldNatFree nat ma = do (a, s) <- get >>= nat . S.runStateT ma put s return a codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'L.WriterT'@ monad is the class of all writer -- monads. type instance AlgebraType0 (L.WriterT w) m = ( Monad m, Monoid w ) type instance AlgebraType (L.WriterT w) m = ( MonadWriter w m ) -- | -- Lazy @'L.WriterT'@ is free for algebras of type @'MonadWriter'@. instance FreeAlgebra1 (L.WriterT w) where liftFree = lift foldNatFree nat (L.WriterT m) = fst <$> nat m codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'S.WriterT'@ monad is the class of all writer -- monads. type instance AlgebraType0 (S.WriterT w) m = ( Monad m, Monoid w ) type instance AlgebraType (S.WriterT w) m = ( MonadWriter w m ) -- | -- Strict @'S.WriterT'@ monad transformer is a free algebra among all -- @'MonadWriter'@s. instance FreeAlgebra1 (S.WriterT w) where liftFree = lift foldNatFree nat (S.WriterT m) = fst <$> nat m codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'L.ReaderT'@ monad is the class of all reader -- monads. -- -- TODO: take advantage of poly-kinded `ReaderT` type instance AlgebraType0 (ReaderT r) m = ( Monad m ) type instance AlgebraType (ReaderT r) m = ( MonadReader r m ) -- | -- @'ReaderT'@ is a free monad in the class of all @'MonadReader'@ monads. instance FreeAlgebra1 (ReaderT r :: (Type -> Type) -> Type -> Type) where liftFree = lift foldNatFree nat (ReaderT g) = ask >>= nat . g codom1 = proof forget1 = proof -- | -- Algebras of the same type as @'S.ReaderT'@ monad is the class of all reader -- monads. type instance AlgebraType0 (ExceptT e) m = ( Monad m ) type instance AlgebraType (ExceptT e) m = ( MonadError e m ) -- | -- @'ExceptT' e@ is a free algebra among all @'MonadError' e@ monads. instance FreeAlgebra1 (ExceptT e) where liftFree = lift foldNatFree nat (ExceptT m) = do ea <- nat m case ea of Left e -> throwError e Right a -> return a codom1 = proof forget1 = proof type instance AlgebraType0 (L.RWST r w s) m = ( Monad m, Monoid w ) type instance AlgebraType (L.RWST r w s) m = MonadRWS r w s m instance FreeAlgebra1 (L.RWST r w s) where liftFree = lift foldNatFree nat (L.RWST fn) = do r <- ask s <- get (a, s', w) <- nat $ fn r s put s' tell w return a codom1 = proof forget1 = proof type instance AlgebraType0 (S.RWST r w s) m = ( Monad m, Monoid w ) type instance AlgebraType (S.RWST r w s) m = MonadRWS r w s m instance FreeAlgebra1 (S.RWST r w s) where liftFree = lift foldNatFree nat (S.RWST fn) = do r <- ask s <- get (a, s', w) <- nat $ fn r s put s' tell w return a codom1 = proof forget1 = proof -- | -- Algebra type for @'ListT'@ monad transformer. class Monad m => MonadList m where mempty1 :: m a mappend1 :: m a -> m a -> m a mappend1_ :: MonadList m => a -> a -> m a mappend1_ a b = return a `mappend1` return b {-# INLINE mappend1_ #-} instance Monad m => MonadList (ListT m) where mempty1 = ListT (return []) mappend1 (ListT ma) (ListT mb) = ListT $ mappend <$> ma <*> mb type instance AlgebraType0 ListT f = ( Monad f ) type instance AlgebraType ListT m = ( MonadList m ) instance FreeAlgebra1 ListT where liftFree = lift foldNatFree nat (ListT mas) = do as <- nat mas empty <- mempty1 a <- foldM (\x y -> x `mappend1_` y) empty as return a codom1 = proof forget1 = proof -- $monadContT -- -- @'ContT' r m@ is not functorial in @m@, so there is no chance it can admit -- an instance of @'FreeAlgebra1'@ -- | -- A higher version @'Data.Algebra.Pointed'@ class. -- -- With @'QuantifiedConstraints'@ this class will be redundant. class MonadMaybe m where point :: forall a. m a instance Monad m => MonadMaybe (MaybeT m) where point = MaybeT (return Nothing) type instance AlgebraType0 MaybeT m = ( Monad m ) type instance AlgebraType MaybeT m = ( Monad m, MonadMaybe m ) instance FreeAlgebra1 MaybeT where liftFree = lift foldNatFree nat (MaybeT mma) = nat mma >>= \ma -> case ma of Nothing -> point Just a -> return a codom1 = proof forget1 = proof