{-| A monad morphism is a natural transformation: > morph :: forall a . m a -> n a ... that obeys the following two laws: > morph $ do x <- m = do x <- morph m > f x morph (f x) > > morph (return x) = return x ... which are equivalent to the following two functor laws: > morph . (f >=> g) = morph . f >=> morph . g > > morph . return = return Examples of monad morphisms include: * 'lift' (from 'MonadTrans') * 'squash' (See below) * @'hoist' f@ (See below), if @f@ is a monad morphism * @(f . g)@, if @f@ and @g@ are both monad morphisms * 'id' Monad morphisms commonly arise when manipulating existing monad transformer code for compatibility purposes. The 'MFunctor', 'MonadTrans', and 'MMonad' classes define standard ways to change monad transformer stacks: * 'lift' introduces a new monad transformer layer of any type. * 'squash' flattens two identical monad transformer layers into a single layer of the same type. * 'hoist' maps monad morphisms to modify deeper layers of the monad transformer stack. -} {-# LANGUAGE Rank2Types #-} module Control.Monad.Morph ( -- * Functors over Monads MFunctor(..), generalize, -- * Monads over Monads MMonad(..), MonadTrans(lift), squash, (>|>), (<|<), (=<|), (|>=) -- * Tutorial -- $tutorial -- ** Generalizing base monads -- $generalize -- ** Monad morphisms -- $mmorph -- ** Mixing diverse transformers -- $interleave -- ** Embedding transformers -- $embed ) where import Control.Applicative.Lift (Lift (Pure, Other)) import Control.Applicative.Backwards (Backwards (Backwards)) import Control.Monad.Trans.Class (MonadTrans(lift)) import qualified Control.Monad.Trans.Error as E import qualified Control.Monad.Trans.Identity as I import qualified Control.Monad.Trans.List as L import qualified Control.Monad.Trans.Maybe as M import qualified Control.Monad.Trans.Reader as R import qualified Control.Monad.Trans.RWS.Lazy as RWS import qualified Control.Monad.Trans.RWS.Strict as RWS' import qualified Control.Monad.Trans.State.Lazy as S import qualified Control.Monad.Trans.State.Strict as S' import qualified Control.Monad.Trans.Writer.Lazy as W' import qualified Control.Monad.Trans.Writer.Strict as W import Data.Monoid (Monoid, mappend) import Data.Functor.Compose (Compose (Compose)) import Data.Functor.Identity (runIdentity) import Data.Functor.Product (Product (Pair)) -- For documentation import Control.Exception (try, IOException) import Control.Monad ((=<<), (>=>), (<=<), join) import Data.Functor.Identity (Identity) {-| A functor in the category of monads, using 'hoist' as the analog of 'fmap': > hoist (f . g) = hoist f . hoist g > > hoist id = id -} class MFunctor t where {-| Lift a monad morphism from @m@ to @n@ into a monad morphism from @(t m)@ to @(t n)@ -} hoist :: (Monad m) => (forall a . m a -> n a) -> t m b -> t n b instance MFunctor (E.ErrorT e) where hoist nat m = E.ErrorT (nat (E.runErrorT m)) instance MFunctor I.IdentityT where hoist nat m = I.IdentityT (nat (I.runIdentityT m)) instance MFunctor L.ListT where hoist nat m = L.ListT (nat (L.runListT m)) instance MFunctor M.MaybeT where hoist nat m = M.MaybeT (nat (M.runMaybeT m)) instance MFunctor (R.ReaderT r) where hoist nat m = R.ReaderT (\i -> nat (R.runReaderT m i)) instance MFunctor (RWS.RWST r w s) where hoist nat m = RWS.RWST (\r s -> nat (RWS.runRWST m r s)) instance MFunctor (RWS'.RWST r w s) where hoist nat m = RWS'.RWST (\r s -> nat (RWS'.runRWST m r s)) instance MFunctor (S.StateT s) where hoist nat m = S.StateT (\s -> nat (S.runStateT m s)) instance MFunctor (S'.StateT s) where hoist nat m = S'.StateT (\s -> nat (S'.runStateT m s)) instance MFunctor (W.WriterT w) where hoist nat m = W.WriterT (nat (W.runWriterT m)) instance MFunctor (W'.WriterT w) where hoist nat m = W'.WriterT (nat (W'.runWriterT m)) instance Functor f => MFunctor (Compose f) where hoist nat (Compose f) = Compose (fmap nat f) instance MFunctor (Product f) where hoist nat (Pair f g) = Pair f (nat g) instance MFunctor Backwards where hoist nat (Backwards f) = Backwards (nat f) instance MFunctor Lift where hoist _ (Pure a) = Pure a hoist nat (Other f) = Other (nat f) -- | A function that @generalize@s the 'Identity' base monad to be any monad. generalize :: Monad m => Identity a -> m a generalize = return . runIdentity {-# INLINABLE generalize #-} {-| A monad in the category of monads, using 'lift' from 'MonadTrans' as the analog of 'return' and 'embed' as the analog of ('=<<'): > embed lift = id > > embed f (lift m) = f m > > embed g (embed f t) = embed (\m -> embed g (f m)) t -} class (MFunctor t, MonadTrans t) => MMonad t where {-| Embed a newly created 'MMonad' layer within an existing layer 'embed' is analogous to ('=<<') -} embed :: (Monad n) => (forall a . m a -> t n a) -> t m b -> t n b {-| Squash two 'MMonad' layers into a single layer 'squash' is analogous to 'join' -} squash :: (Monad m, MMonad t) => t (t m) a -> t m a squash = embed id {-# INLINABLE squash #-} infixr 2 >|>, =<| infixl 2 <|<, |>= {-| Compose two 'MMonad' layer-building functions ('>|>') is analogous to ('>=>') -} (>|>) :: (Monad m3, MMonad t) => (forall a . m1 a -> t m2 a) -> (forall b . m2 b -> t m3 b) -> m1 c -> t m3 c (f >|> g) m = embed g (f m) {-# INLINABLE (>|>) #-} {-| Equivalent to ('>|>') with the arguments flipped ('<|<') is analogous to ('<=<') -} (<|<) :: (Monad m3, MMonad t) => (forall b . m2 b -> t m3 b) -> (forall a . m1 a -> t m2 a) -> m1 c -> t m3 c (g <|< f) m = embed g (f m) {-# INLINABLE (<|<) #-} {-| An infix operator equivalent to 'embed' ('=<|') is analogous to ('=<<') -} (=<|) :: (Monad n, MMonad t) => (forall a . m a -> t n a) -> t m b -> t n b (=<|) = embed {-# INLINABLE (=<|) #-} {-| Equivalent to ('=<|') with the arguments flipped ('|>=') is analogous to ('>>=') -} (|>=) :: (Monad n, MMonad t) => t m b -> (forall a . m a -> t n a) -> t n b t |>= f = embed f t {-# INLINABLE (|>=) #-} instance (E.Error e) => MMonad (E.ErrorT e) where embed f m = E.ErrorT (do x <- E.runErrorT (f (E.runErrorT m)) return (case x of Left e -> Left e Right (Left e) -> Left e Right (Right a) -> Right a ) ) instance MMonad I.IdentityT where embed f m = f (I.runIdentityT m) instance MMonad L.ListT where embed f m = L.ListT (do x <- L.runListT (f (L.runListT m)) return (concat x)) instance MMonad M.MaybeT where embed f m = M.MaybeT (do x <- M.runMaybeT (f (M.runMaybeT m)) return (case x of Nothing -> Nothing Just Nothing -> Nothing Just (Just a) -> Just a ) ) instance MMonad (R.ReaderT r) where embed f m = R.ReaderT (\i -> R.runReaderT (f (R.runReaderT m i)) i) instance (Monoid w) => MMonad (W.WriterT w) where embed f m = W.WriterT (do ~((a, w1), w2) <- W.runWriterT (f (W.runWriterT m)) return (a, mappend w1 w2) ) instance (Monoid w) => MMonad (W'.WriterT w) where embed f m = W'.WriterT (do ((a, w1), w2) <- W'.runWriterT (f (W'.runWriterT m)) return (a, mappend w1 w2) ) {- $tutorial Monad morphisms solve the common problem of fixing monadic code after the fact without modifying the original source code or type signatures. The following sections illustrate various examples of transparently modifying existing functions. -} {- $generalize Imagine that some library provided the following 'S.State' code: > import Control.Monad.Trans.State > > tick :: State Int () > tick = modify (+1) ... but we would prefer to reuse @tick@ within a larger @('S.StateT' Int 'IO')@ block in order to mix in 'IO' actions. We could patch the original library to generalize @tick@'s type signature: > tick :: (Monad m) => StateT Int m () ... but we would prefer not to fork upstream code if possible. How could we generalize @tick@'s type without modifying the original code? We can solve this if we realize that 'S.State' is a type synonym for 'S.StateT' with an 'Identity' base monad: > type State s = StateT s Identity ... which means that @tick@'s true type is actually: > tick :: StateT Int Identity () Now all we need is a function that @generalize@s the 'Identity' base monad to be any monad: > import Data.Functor.Identity > > generalize :: (Monad m) => Identity a -> m a > generalize m = return (runIdentity m) ... which we can 'hoist' to change @tick@'s base monad: > hoist :: (Monad m, MFunctor t) => (forall a . m a -> n a) -> t m b -> t n b > > hoist generalize :: (Monad m, MFunctor t) => t Identity b -> t m b > > hoist generalize tick :: (Monad m) => StateT Int m () This lets us mix @tick@ alongside 'IO' using 'lift': > import Control.Monad.Morph > import Control.Monad.Trans.Class > > tock :: StateT Int IO () > tock = do > hoist generalize tick :: (Monad m) => StateT Int m () > lift $ putStrLn "Tock!" :: (MonadTrans t) => t IO () >>> runStateT tock 0 Tock! ((), 1) -} {- $mmorph Notice that @generalize@ is a monad morphism, and the following two proofs show how @generalize@ satisfies the monad morphism laws. You can refer to these proofs as an example for how to prove a function obeys the monad morphism laws: > generalize (return x) > > -- Definition of 'return' for the Identity monad > = generalize (Identity x) > > -- Definition of 'generalize' > = return (runIdentity (Identity x)) > > -- runIdentity (Identity x) = x > = return x > generalize $ do x <- m > f x > > -- Definition of (>>=) for the Identity monad > = generalize (f (runIdentity m)) > > -- Definition of 'generalize' > = return (runIdentity (f (runIdentity m))) > > -- Monad law: Left identity > = do x <- return (runIdentity m) > return (runIdentity (f x)) > > -- Definition of 'generalize' in reverse > = do x <- generalize m > generalize (f x) -} {- $interleave You can combine 'hoist' and 'lift' to insert arbitrary layers anywhere within a monad transformer stack. This comes in handy when interleaving two diverse stacks. For example, we might want to combine the following @save@ function: > import Control.Monad.Trans.Writer > > -- i.e. :: StateT Int (WriterT [Int] Identity) () > save :: StateT Int (Writer [Int]) () > save = do > n <- get > lift $ tell [n] ... with our previous @tock@ function: > tock :: StateT Int IO () However, @save@ and @tock@ differ in two ways: * @tock@ lacks a 'W.WriterT' layer * @save@ has an 'Identity' base monad We can mix the two by inserting a 'W.WriterT' layer for @tock@ and generalizing @save@'s base monad: > import Control.Monad > > program :: StateT Int (WriterT [Int] IO) () > program = replicateM_ 4 $ do > hoist lift tock > :: (MonadTrans t) => StateT Int (t IO) () > hoist (hoist generalize) save > :: (Monad m) => StateT Int (WriterT [Int] m ) () >>> execWriterT (runStateT program 0) Tock! Tock! Tock! Tock! [1,2,3,4] -} {- $embed Suppose we decided to @check@ all 'IOException's using a combination of 'try' and 'ErrorT': > import Control.Exception > import Control.Monad.Trans.Class > import Control.Monad.Trans.Error > > check :: IO a -> ErrorT IOException IO a > check io = ErrorT (try io) ... but then we forget to use @check@ in one spot, mistakenly using 'lift' instead: > program :: ErrorT IOException IO () > program = do > str <- lift $ readFile "test.txt" > check $ putStr str >>> runErrorT program *** Exception: test.txt: openFile: does not exist (No such file or directory) How could we go back and fix 'program' without modifying its source code? Well, @check@ is a monad morphism, but we can't 'hoist' it to modify the base monad because then we get two 'E.ErrorT' layers instead of one: > hoist check :: (MFunctor t) => t IO a -> t (ErrorT IOException IO) a > > hoist check program :: ErrorT IOException (ErrorT IOException IO) () We'd prefer to 'embed' all newly generated exceptions in the existing 'E.ErrorT' layer: > embed check :: ErrorT IOException IO a -> ErrorT IOException IO a > > embed check program :: ErrorT IOException IO () This correctly checks the exceptions that slipped through the cracks: >>> import Control.Monad.Morph >>> runErrorT (embed check program) Left test.txt: openFile: does not exist (No such file or directory) -}