content: {-# LANGUAGE CPP        #-}
  {-# LANGUAGE RankNTypes #-}
  {-# LANGUAGE Safe       #-}
  #if __GLASGOW_HASKELL__ >= 706
  {-# LANGUAGE PolyKinds  #-}
  #endif
  
  {-| 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.
  
  -}
  
  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.Monad.Trans.Class (MonadTrans(lift))
  import qualified Control.Monad.Trans.Accum         as A
  import qualified Control.Monad.Trans.Except        as Ex
  import qualified Control.Monad.Trans.Identity      as I
  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))
  import Control.Applicative.Backwards (Backwards (Backwards))
  import Control.Applicative.Lift (Lift (Pure, Other))
  
  -- 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)@
  
          The first argument to `hoist` must be a monad morphism, even though the
          type system does not enforce this
      -}
      hoist :: (Monad m) => (forall a . m a -> n a) -> t m b -> t n b
  
  instance MFunctor (A.AccumT w) where
      hoist nat m = A.AccumT (nat . A.runAccumT m)
  
  instance MFunctor (Ex.ExceptT e) where
      hoist nat m = Ex.ExceptT (nat (Ex.runExceptT m))
  
  instance MFunctor I.IdentityT where
      hoist nat m = I.IdentityT (nat (I.runIdentityT 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 Monoid w => MMonad (A.AccumT w) where
      embed f m = A.AccumT $ \w -> do
          ((b, wInner), wOuter) <- A.runAccumT (f $ A.runAccumT m w) w
          return (b, wInner `mappend` wOuter)
  
  instance MMonad (Ex.ExceptT e) where
      embed f m = Ex.ExceptT (do
          x <- Ex.runExceptT (f (Ex.runExceptT 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 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)
  
  -}
  