{-# 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.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 (Ex.ExceptT e) where
    hoist :: (forall a. m a -> n a) -> ExceptT e m b -> ExceptT e n b
hoist forall a. m a -> n a
nat ExceptT e m b
m = n (Either e b) -> ExceptT e n b
forall e (m :: * -> *) a. m (Either e a) -> ExceptT e m a
Ex.ExceptT (m (Either e b) -> n (Either e b)
forall a. m a -> n a
nat (ExceptT e m b -> m (Either e b)
forall e (m :: * -> *) a. ExceptT e m a -> m (Either e a)
Ex.runExceptT ExceptT e m b
m))

instance MFunctor I.IdentityT where
    hoist :: (forall a. m a -> n a) -> IdentityT m b -> IdentityT n b
hoist forall a. m a -> n a
nat IdentityT m b
m = n b -> IdentityT n b
forall k (f :: k -> *) (a :: k). f a -> IdentityT f a
I.IdentityT (m b -> n b
forall a. m a -> n a
nat (IdentityT m b -> m b
forall k (f :: k -> *) (a :: k). IdentityT f a -> f a
I.runIdentityT IdentityT m b
m))

instance MFunctor M.MaybeT where
    hoist :: (forall a. m a -> n a) -> MaybeT m b -> MaybeT n b
hoist forall a. m a -> n a
nat MaybeT m b
m = n (Maybe b) -> MaybeT n b
forall (m :: * -> *) a. m (Maybe a) -> MaybeT m a
M.MaybeT (m (Maybe b) -> n (Maybe b)
forall a. m a -> n a
nat (MaybeT m b -> m (Maybe b)
forall (m :: * -> *) a. MaybeT m a -> m (Maybe a)
M.runMaybeT MaybeT m b
m))

instance MFunctor (R.ReaderT r) where
    hoist :: (forall a. m a -> n a) -> ReaderT r m b -> ReaderT r n b
hoist forall a. m a -> n a
nat ReaderT r m b
m = (r -> n b) -> ReaderT r n b
forall r (m :: * -> *) a. (r -> m a) -> ReaderT r m a
R.ReaderT (\r
i -> m b -> n b
forall a. m a -> n a
nat (ReaderT r m b -> r -> m b
forall r (m :: * -> *) a. ReaderT r m a -> r -> m a
R.runReaderT ReaderT r m b
m r
i))

instance MFunctor (RWS.RWST r w s) where
    hoist :: (forall a. m a -> n a) -> RWST r w s m b -> RWST r w s n b
hoist forall a. m a -> n a
nat RWST r w s m b
m = (r -> s -> n (b, s, w)) -> RWST r w s n b
forall r w s (m :: * -> *) a.
(r -> s -> m (a, s, w)) -> RWST r w s m a
RWS.RWST (\r
r s
s -> m (b, s, w) -> n (b, s, w)
forall a. m a -> n a
nat (RWST r w s m b -> r -> s -> m (b, s, w)
forall r w s (m :: * -> *) a.
RWST r w s m a -> r -> s -> m (a, s, w)
RWS.runRWST RWST r w s m b
m r
r s
s))

instance MFunctor (RWS'.RWST r w s) where
    hoist :: (forall a. m a -> n a) -> RWST r w s m b -> RWST r w s n b
hoist forall a. m a -> n a
nat RWST r w s m b
m = (r -> s -> n (b, s, w)) -> RWST r w s n b
forall r w s (m :: * -> *) a.
(r -> s -> m (a, s, w)) -> RWST r w s m a
RWS'.RWST (\r
r s
s -> m (b, s, w) -> n (b, s, w)
forall a. m a -> n a
nat (RWST r w s m b -> r -> s -> m (b, s, w)
forall r w s (m :: * -> *) a.
RWST r w s m a -> r -> s -> m (a, s, w)
RWS'.runRWST RWST r w s m b
m r
r s
s))

instance MFunctor (S.StateT s) where
    hoist :: (forall a. m a -> n a) -> StateT s m b -> StateT s n b
hoist forall a. m a -> n a
nat StateT s m b
m = (s -> n (b, s)) -> StateT s n b
forall s (m :: * -> *) a. (s -> m (a, s)) -> StateT s m a
S.StateT (\s
s -> m (b, s) -> n (b, s)
forall a. m a -> n a
nat (StateT s m b -> s -> m (b, s)
forall s (m :: * -> *) a. StateT s m a -> s -> m (a, s)
S.runStateT StateT s m b
m s
s))

instance MFunctor (S'.StateT s) where
    hoist :: (forall a. m a -> n a) -> StateT s m b -> StateT s n b
hoist forall a. m a -> n a
nat StateT s m b
m = (s -> n (b, s)) -> StateT s n b
forall s (m :: * -> *) a. (s -> m (a, s)) -> StateT s m a
S'.StateT (\s
s -> m (b, s) -> n (b, s)
forall a. m a -> n a
nat (StateT s m b -> s -> m (b, s)
forall s (m :: * -> *) a. StateT s m a -> s -> m (a, s)
S'.runStateT StateT s m b
m s
s))

instance MFunctor (W.WriterT w) where
    hoist :: (forall a. m a -> n a) -> WriterT w m b -> WriterT w n b
hoist forall a. m a -> n a
nat WriterT w m b
m = n (b, w) -> WriterT w n b
forall w (m :: * -> *) a. m (a, w) -> WriterT w m a
W.WriterT (m (b, w) -> n (b, w)
forall a. m a -> n a
nat (WriterT w m b -> m (b, w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W.runWriterT WriterT w m b
m))

instance MFunctor (W'.WriterT w) where
    hoist :: (forall a. m a -> n a) -> WriterT w m b -> WriterT w n b
hoist forall a. m a -> n a
nat WriterT w m b
m = n (b, w) -> WriterT w n b
forall w (m :: * -> *) a. m (a, w) -> WriterT w m a
W'.WriterT (m (b, w) -> n (b, w)
forall a. m a -> n a
nat (WriterT w m b -> m (b, w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W'.runWriterT WriterT w m b
m))

instance Functor f => MFunctor (Compose f) where
    hoist :: (forall a. m a -> n a) -> Compose f m b -> Compose f n b
hoist forall a. m a -> n a
nat (Compose f (m b)
f) = f (n b) -> Compose f n b
forall k k1 (f :: k -> *) (g :: k1 -> k) (a :: k1).
f (g a) -> Compose f g a
Compose ((m b -> n b) -> f (m b) -> f (n b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap m b -> n b
forall a. m a -> n a
nat f (m b)
f)

instance MFunctor (Product f) where
    hoist :: (forall a. m a -> n a) -> Product f m b -> Product f n b
hoist forall a. m a -> n a
nat (Pair f b
f m b
g) = f b -> n b -> Product f n b
forall k (f :: k -> *) (g :: k -> *) (a :: k).
f a -> g a -> Product f g a
Pair f b
f (m b -> n b
forall a. m a -> n a
nat m b
g)

instance MFunctor Backwards where
    hoist :: (forall a. m a -> n a) -> Backwards m b -> Backwards n b
hoist forall a. m a -> n a
nat (Backwards m b
f) = n b -> Backwards n b
forall k (f :: k -> *) (a :: k). f a -> Backwards f a
Backwards (m b -> n b
forall a. m a -> n a
nat m b
f)

instance MFunctor Lift where
    hoist :: (forall a. m a -> n a) -> Lift m b -> Lift n b
hoist forall a. m a -> n a
_   (Pure b
a)  = b -> Lift n b
forall (f :: * -> *) a. a -> Lift f a
Pure b
a
    hoist forall a. m a -> n a
nat (Other m b
f) = n b -> Lift n b
forall (f :: * -> *) a. f a -> Lift f a
Other (m b -> n b
forall a. m a -> n a
nat m b
f)

-- | A function that @generalize@s the 'Identity' base monad to be any monad.
generalize :: Monad m => Identity a -> m a
generalize :: Identity a -> m a
generalize = a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (a -> m a) -> (Identity a -> a) -> Identity a -> m a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Identity a -> a
forall a. Identity a -> a
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 :: t (t m) a -> t m a
squash = (forall a. t m a -> t m a) -> t (t m) a -> t m a
forall (t :: (* -> *) -> * -> *) (n :: * -> *) (m :: * -> *) b.
(MMonad t, Monad n) =>
(forall a. m a -> t n a) -> t m b -> t n b
embed forall a. a -> a
forall a. t m a -> t m a
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
(forall a. m1 a -> t m2 a
f >|> :: (forall a. m1 a -> t m2 a)
-> (forall b. m2 b -> t m3 b) -> m1 c -> t m3 c
>|> forall b. m2 b -> t m3 b
g) m1 c
m = (forall b. m2 b -> t m3 b) -> t m2 c -> t m3 c
forall (t :: (* -> *) -> * -> *) (n :: * -> *) (m :: * -> *) b.
(MMonad t, Monad n) =>
(forall a. m a -> t n a) -> t m b -> t n b
embed forall b. m2 b -> t m3 b
g (m1 c -> t m2 c
forall a. m1 a -> t m2 a
f m1 c
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
(forall b. m2 b -> t m3 b
g <|< :: (forall b. m2 b -> t m3 b)
-> (forall a. m1 a -> t m2 a) -> m1 c -> t m3 c
<|< forall a. m1 a -> t m2 a
f) m1 c
m = (forall b. m2 b -> t m3 b) -> t m2 c -> t m3 c
forall (t :: (* -> *) -> * -> *) (n :: * -> *) (m :: * -> *) b.
(MMonad t, Monad n) =>
(forall a. m a -> t n a) -> t m b -> t n b
embed forall b. m2 b -> t m3 b
g (m1 c -> t m2 c
forall a. m1 a -> t m2 a
f m1 c
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
=<| :: (forall a. m a -> t n a) -> t m b -> t n b
(=<|) = (forall a. m a -> t n a) -> t m b -> t n b
forall (t :: (* -> *) -> * -> *) (n :: * -> *) (m :: * -> *) b.
(MMonad t, Monad n) =>
(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 m b
t |>= :: t m b -> (forall a. m a -> t n a) -> t n b
|>= forall a. m a -> t n a
f = (forall a. m a -> t n a) -> t m b -> t n b
forall (t :: (* -> *) -> * -> *) (n :: * -> *) (m :: * -> *) b.
(MMonad t, Monad n) =>
(forall a. m a -> t n a) -> t m b -> t n b
embed forall a. m a -> t n a
f t m b
t
{-# INLINABLE (|>=) #-}

instance MMonad (Ex.ExceptT e) where
    embed :: (forall a. m a -> ExceptT e n a) -> ExceptT e m b -> ExceptT e n b
embed forall a. m a -> ExceptT e n a
f ExceptT e m b
m = n (Either e b) -> ExceptT e n b
forall e (m :: * -> *) a. m (Either e a) -> ExceptT e m a
Ex.ExceptT (do
        Either e (Either e b)
x <- ExceptT e n (Either e b) -> n (Either e (Either e b))
forall e (m :: * -> *) a. ExceptT e m a -> m (Either e a)
Ex.runExceptT (m (Either e b) -> ExceptT e n (Either e b)
forall a. m a -> ExceptT e n a
f (ExceptT e m b -> m (Either e b)
forall e (m :: * -> *) a. ExceptT e m a -> m (Either e a)
Ex.runExceptT ExceptT e m b
m))
        Either e b -> n (Either e b)
forall (m :: * -> *) a. Monad m => a -> m a
return (case Either e (Either e b)
x of
            Left         e
e  -> e -> Either e b
forall a b. a -> Either a b
Left e
e
            Right (Left  e
e) -> e -> Either e b
forall a b. a -> Either a b
Left e
e
            Right (Right b
a) -> b -> Either e b
forall a b. b -> Either a b
Right b
a ) )

instance MMonad I.IdentityT where
    embed :: (forall a. m a -> IdentityT n a) -> IdentityT m b -> IdentityT n b
embed forall a. m a -> IdentityT n a
f IdentityT m b
m = m b -> IdentityT n b
forall a. m a -> IdentityT n a
f (IdentityT m b -> m b
forall k (f :: k -> *) (a :: k). IdentityT f a -> f a
I.runIdentityT IdentityT m b
m)

instance MMonad M.MaybeT where
    embed :: (forall a. m a -> MaybeT n a) -> MaybeT m b -> MaybeT n b
embed forall a. m a -> MaybeT n a
f MaybeT m b
m = n (Maybe b) -> MaybeT n b
forall (m :: * -> *) a. m (Maybe a) -> MaybeT m a
M.MaybeT (do
        Maybe (Maybe b)
x <- MaybeT n (Maybe b) -> n (Maybe (Maybe b))
forall (m :: * -> *) a. MaybeT m a -> m (Maybe a)
M.runMaybeT (m (Maybe b) -> MaybeT n (Maybe b)
forall a. m a -> MaybeT n a
f (MaybeT m b -> m (Maybe b)
forall (m :: * -> *) a. MaybeT m a -> m (Maybe a)
M.runMaybeT MaybeT m b
m))
        Maybe b -> n (Maybe b)
forall (m :: * -> *) a. Monad m => a -> m a
return (case Maybe (Maybe b)
x of
            Maybe (Maybe b)
Nothing       -> Maybe b
forall a. Maybe a
Nothing
            Just Maybe b
Nothing  -> Maybe b
forall a. Maybe a
Nothing
            Just (Just b
a) -> b -> Maybe b
forall a. a -> Maybe a
Just b
a ) )

instance MMonad (R.ReaderT r) where
    embed :: (forall a. m a -> ReaderT r n a) -> ReaderT r m b -> ReaderT r n b
embed forall a. m a -> ReaderT r n a
f ReaderT r m b
m = (r -> n b) -> ReaderT r n b
forall r (m :: * -> *) a. (r -> m a) -> ReaderT r m a
R.ReaderT (\r
i -> ReaderT r n b -> r -> n b
forall r (m :: * -> *) a. ReaderT r m a -> r -> m a
R.runReaderT (m b -> ReaderT r n b
forall a. m a -> ReaderT r n a
f (ReaderT r m b -> r -> m b
forall r (m :: * -> *) a. ReaderT r m a -> r -> m a
R.runReaderT ReaderT r m b
m r
i)) r
i)

instance (Monoid w) => MMonad (W.WriterT w) where
    embed :: (forall a. m a -> WriterT w n a) -> WriterT w m b -> WriterT w n b
embed forall a. m a -> WriterT w n a
f WriterT w m b
m = n (b, w) -> WriterT w n b
forall w (m :: * -> *) a. m (a, w) -> WriterT w m a
W.WriterT (do
        ~((b
a, w
w1), w
w2) <- WriterT w n (b, w) -> n ((b, w), w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W.runWriterT (m (b, w) -> WriterT w n (b, w)
forall a. m a -> WriterT w n a
f (WriterT w m b -> m (b, w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W.runWriterT WriterT w m b
m))
        (b, w) -> n (b, w)
forall (m :: * -> *) a. Monad m => a -> m a
return (b
a, w -> w -> w
forall a. Monoid a => a -> a -> a
mappend w
w1 w
w2) )

instance (Monoid w) => MMonad (W'.WriterT w) where
    embed :: (forall a. m a -> WriterT w n a) -> WriterT w m b -> WriterT w n b
embed forall a. m a -> WriterT w n a
f WriterT w m b
m = n (b, w) -> WriterT w n b
forall w (m :: * -> *) a. m (a, w) -> WriterT w m a
W'.WriterT (do
        ((b
a, w
w1), w
w2) <- WriterT w n (b, w) -> n ((b, w), w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W'.runWriterT (m (b, w) -> WriterT w n (b, w)
forall a. m a -> WriterT w n a
f (WriterT w m b -> m (b, w)
forall w (m :: * -> *) a. WriterT w m a -> m (a, w)
W'.runWriterT WriterT w m b
m))
        (b, w) -> n (b, w)
forall (m :: * -> *) a. Monad m => a -> m a
return (b
a, w -> w -> w
forall a. Monoid a => a -> a -> a
mappend w
w1 w
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)

-}