{-# LANGUAGE CPP #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE Rank2Types #-}
#if __GLASGOW_HASKELL__ >= 707
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE Safe #-}
#else
-- Manual Typeable instances

{-# LANGUAGE Trustworthy #-}
#endif
#include "free-common.h"

-----------------------------------------------------------------------------

-- |

-- Module      :  Control.Monad.Free

-- Copyright   :  (C) 2008-2015 Edward Kmett

-- License     :  BSD-style (see the file LICENSE)

--

-- Maintainer  :  Edward Kmett <ekmett@gmail.com>

-- Stability   :  provisional

-- Portability :  MPTCs, fundeps

--

-- Monads for free

----------------------------------------------------------------------------

module Control.Monad.Free
  ( 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 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.Compat
import Data.Functor.WithIndex
import Data.Foldable
import Data.Foldable.WithIndex
import Data.Profunctor
import Data.Traversable
import Data.Traversable.WithIndex
import Data.Semigroup.Foldable
import Data.Semigroup.Traversable
import Data.Data
import Prelude hiding (foldr)
#if __GLASGOW_HASKELL__ >= 707
import GHC.Generics
#endif

-- $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))))


-- | The 'Free' 'Monad' for a 'Functor' @f@.

--

-- /Formally/

--

-- A 'Monad' @n@ is a free 'Monad' for @f@ if every monad homomorphism

-- from @n@ to another monad @m@ is equivalent to a natural transformation

-- from @f@ to @m@.

--

-- /Why Free?/

--

-- Every \"free\" functor is left adjoint to some \"forgetful\" functor.

--

-- If we define a forgetful functor @U@ from the category of monads to the category of functors

-- that just forgets the 'Monad', leaving only the 'Functor'. i.e.

--

-- @U (M,'return','Control.Monad.join') = M@

--

-- then 'Free' is the left adjoint to @U@.

--

-- 'Free' being left adjoint to @U@ means that there is an isomorphism between

--

-- @'Free' f -> m@ in the category of monads and @f -> U m@ in the category of functors.

--

-- Morphisms in the category of monads are 'Monad' homomorphisms (natural transformations that respect 'return' and 'Control.Monad.join').

--

-- Morphisms in the category of functors are 'Functor' homomorphisms (natural transformations).

--

-- Given this isomorphism, every monad homomorphism from @'Free' f@ to @m@ is equivalent to a natural transformation from @f@ to @m@

--

-- Showing that this isomorphism holds is left as an exercise.

--

-- In practice, you can just view a @'Free' f a@ as many layers of @f@ wrapped around values of type @a@, where

-- @('>>=')@ performs substitution and grafts new layers of @f@ in for each of the free variables.

--

-- This can be very useful for modeling domain specific languages, trees, or other constructs.

--

-- This instance of 'MonadFree' is fairly naive about the encoding. For more efficient free monad implementation see "Control.Monad.Free.Church", in particular note the 'Control.Monad.Free.Church.improve' combinator.

-- You may also want to take a look at the @kan-extensions@ package (<http://hackage.haskell.org/package/kan-extensions>).

--

-- A number of common monads arise as free monads,

--

-- * Given @data Empty a@, @'Free' Empty@ is isomorphic to the 'Data.Functor.Identity' monad.

--

-- * @'Free' 'Maybe'@ can be used to model a partiality monad where each layer represents running the computation for a while longer.

data Free f a = Pure a | Free (f (Free f a))
#if __GLASGOW_HASKELL__ >= 707
  deriving (Typeable, forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall (f :: * -> *) a x. Rep (Free f a) x -> Free f a
forall (f :: * -> *) a x. Free f a -> Rep (Free f a) x
$cto :: forall (f :: * -> *) a x. Rep (Free f a) x -> Free f a
$cfrom :: forall (f :: * -> *) a x. Free f a -> Rep (Free f a) x
Generic, forall k (f :: k -> *).
(forall (a :: k). f a -> Rep1 f a)
-> (forall (a :: k). Rep1 f a -> f a) -> Generic1 f
forall (f :: * -> *) a. Functor f => Rep1 (Free f) a -> Free f a
forall (f :: * -> *) a. Functor f => Free f a -> Rep1 (Free f) a
$cto1 :: forall (f :: * -> *) a. Functor f => Rep1 (Free f) a -> Free f a
$cfrom1 :: forall (f :: * -> *) a. Functor f => Free f a -> Rep1 (Free f) a
Generic1)

deriving instance (Typeable f, Data (f (Free f a)), Data a) => Data (Free f a)
#endif

#ifdef LIFTED_FUNCTOR_CLASSES
instance Eq1 f => Eq1 (Free f) where
  liftEq :: forall a b. (a -> b -> Bool) -> Free f a -> Free f b -> Bool
liftEq a -> b -> Bool
eq = forall {f :: * -> *}. Eq1 f => Free f a -> Free f b -> Bool
go
    where
      go :: Free f a -> Free f b -> Bool
go (Pure a
a)  (Pure b
b)  = a -> b -> Bool
eq a
a b
b
      go (Free f (Free f a)
fa) (Free f (Free f b)
fb) = forall (f :: * -> *) a b.
Eq1 f =>
(a -> b -> Bool) -> f a -> f b -> Bool
liftEq Free f a -> Free f b -> Bool
go f (Free f a)
fa f (Free f b)
fb
      go Free f a
_ Free f b
_                 = Bool
False
#else
instance (Functor f, Eq1 f) => Eq1 (Free f) where
  Pure a  `eq1` Pure b  = a == b
  Free fa `eq1` Free fb = fmap Lift1 fa `eq1` fmap Lift1 fb
  _       `eq1` _ = False
#endif

#ifdef LIFTED_FUNCTOR_CLASSES
instance (Eq1 f, Eq a) => Eq (Free f a) where
#else
instance (Eq1 f, Functor f, Eq a) => Eq (Free f a) where
#endif
  == :: Free f a -> Free f a -> Bool
(==) = forall (f :: * -> *) a. (Eq1 f, Eq a) => f a -> f a -> Bool
eq1

#ifdef LIFTED_FUNCTOR_CLASSES
instance Ord1 f => Ord1 (Free f) where
  liftCompare :: forall a b.
(a -> b -> Ordering) -> Free f a -> Free f b -> Ordering
liftCompare a -> b -> Ordering
cmp = forall {f :: * -> *}. Ord1 f => Free f a -> Free f b -> Ordering
go
    where
      go :: Free f a -> Free f b -> Ordering
go (Pure a
a)  (Pure b
b)  = a -> b -> Ordering
cmp a
a b
b
      go (Pure a
_)  (Free f (Free f b)
_)  = Ordering
LT
      go (Free f (Free f a)
_)  (Pure b
_)  = Ordering
GT
      go (Free f (Free f a)
fa) (Free f (Free f b)
fb) = forall (f :: * -> *) a b.
Ord1 f =>
(a -> b -> Ordering) -> f a -> f b -> Ordering
liftCompare Free f a -> Free f b -> Ordering
go f (Free f a)
fa f (Free f b)
fb
#else
instance (Functor f, Ord1 f) => Ord1 (Free f) where
  Pure a `compare1` Pure b = a `compare` b
  Pure _ `compare1` Free _ = LT
  Free _ `compare1` Pure _ = GT
  Free fa `compare1` Free fb = fmap Lift1 fa `compare1` fmap Lift1 fb
#endif

#ifdef LIFTED_FUNCTOR_CLASSES
instance (Ord1 f, Ord a) => Ord (Free f a) where
#else
instance (Ord1 f, Functor f, Ord a) => Ord (Free f a) where
#endif
  compare :: Free f a -> Free f a -> Ordering
compare = forall (f :: * -> *) a. (Ord1 f, Ord a) => f a -> f a -> Ordering
compare1

#ifdef LIFTED_FUNCTOR_CLASSES
instance Show1 f => Show1 (Free f) where
  liftShowsPrec :: forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Free f a -> ShowS
liftShowsPrec Int -> a -> ShowS
sp [a] -> ShowS
sl = forall {f :: * -> *}. Show1 f => Int -> Free f a -> ShowS
go
    where
      go :: Int -> Free f a -> ShowS
go Int
d (Pure a
a) = forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith Int -> a -> ShowS
sp String
"Pure" Int
d a
a
      go Int
d (Free f (Free f a)
fa) = forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith (forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
liftShowsPrec Int -> Free f a -> ShowS
go (forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> [f a] -> ShowS
liftShowList Int -> a -> ShowS
sp [a] -> ShowS
sl)) String
"Free" Int
d f (Free f a)
fa
#else
instance (Functor f, Show1 f) => Show1 (Free f) where
  showsPrec1 d (Pure a) = showParen (d > 10) $
    showString "Pure " . showsPrec 11 a
  showsPrec1 d (Free m) = showParen (d > 10) $
    showString "Free " . showsPrec1 11 (fmap Lift1 m)
#endif

#ifdef LIFTED_FUNCTOR_CLASSES
instance (Show1 f, Show a) => Show (Free f a) where
#else
instance (Show1 f, Functor f, Show a) => Show (Free f a) where
#endif
  showsPrec :: Int -> Free f a -> ShowS
showsPrec = forall (f :: * -> *) a. (Show1 f, Show a) => Int -> f a -> ShowS
showsPrec1

#ifdef LIFTED_FUNCTOR_CLASSES
instance Read1 f => Read1 (Free f) where
  liftReadsPrec :: forall a. (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Free f a)
liftReadsPrec Int -> ReadS a
rp ReadS [a]
rl = Int -> ReadS (Free f a)
go
    where
      go :: Int -> ReadS (Free f a)
go = forall a. (String -> ReadS a) -> Int -> ReadS a
readsData forall a b. (a -> b) -> a -> b
$
        forall a t.
(Int -> ReadS a) -> String -> (a -> t) -> String -> ReadS t
readsUnaryWith Int -> ReadS a
rp String
"Pure" forall (f :: * -> *) a. a -> Free f a
Pure forall a. Monoid a => a -> a -> a
`mappend`
        forall a t.
(Int -> ReadS a) -> String -> (a -> t) -> String -> ReadS t
readsUnaryWith (forall (f :: * -> *) a.
Read1 f =>
(Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (f a)
liftReadsPrec Int -> ReadS (Free f a)
go (forall (f :: * -> *) a.
Read1 f =>
(Int -> ReadS a) -> ReadS [a] -> ReadS [f a]
liftReadList Int -> ReadS a
rp ReadS [a]
rl)) String
"Free" forall (f :: * -> *) a. f (Free f a) -> Free f a
Free
#else
instance (Functor f, Read1 f) => Read1 (Free f) where
  readsPrec1 d r = readParen (d > 10)
      (\r' -> [ (Pure m, t)
             | ("Pure", s) <- lex r'
             , (m, t) <- readsPrec 11 s]) r
    ++ readParen (d > 10)
      (\r' -> [ (Free (fmap lower1 m), t)
             | ("Free", s) <- lex r'
             , (m, t) <- readsPrec1 11 s]) r
#endif

#ifdef LIFTED_FUNCTOR_CLASSES
instance (Read1 f, Read a) => Read (Free f a) where
#else
instance (Read1 f, Functor f, Read a) => Read (Free f a) where
#endif
  readsPrec :: Int -> ReadS (Free f a)
readsPrec = forall (f :: * -> *) a. (Read1 f, Read a) => Int -> ReadS (f a)
readsPrec1

instance Functor f => Functor (Free f) where
  fmap :: forall a b. (a -> b) -> Free f a -> Free f b
fmap a -> b
f = forall {f :: * -> *}. Functor f => Free f a -> Free f b
go where
    go :: Free f a -> Free f b
go (Pure a
a)  = forall (f :: * -> *) a. a -> Free f a
Pure (a -> b
f a
a)
    go (Free f (Free f a)
fa) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (Free f a -> Free f b
go forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
fa)
  {-# INLINE fmap #-}

instance Functor f => Apply (Free f) where
  Pure a -> b
a  <.> :: forall a b. Free f (a -> b) -> Free f a -> Free f b
<.> Pure a
b = forall (f :: * -> *) a. a -> Free f a
Pure (a -> b
a a
b)
  Pure a -> b
a  <.> Free f (Free f a)
fb = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
a forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
fb
  Free f (Free f (a -> b))
fa <.> Free f a
b = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ (forall (f :: * -> *) a b. Apply f => f (a -> b) -> f a -> f b
<.> Free f a
b) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f (a -> b))
fa

instance Functor f => Applicative (Free f) where
  pure :: forall a. a -> Free f a
pure = forall (f :: * -> *) a. a -> Free f a
Pure
  {-# INLINE pure #-}
  Pure a -> b
a <*> :: forall a b. Free f (a -> b) -> Free f a -> Free f b
<*> Pure a
b = forall (f :: * -> *) a. a -> Free f a
Pure forall a b. (a -> b) -> a -> b
$ a -> b
a a
b
  Pure a -> b
a <*> Free f (Free f a)
mb = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
a forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
mb
  Free f (Free f (a -> b))
ma <*> Free f a
b = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ (forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> Free f a
b) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f (a -> b))
ma

instance Functor f => Bind (Free f) where
  Pure a
a >>- :: forall a b. Free f a -> (a -> Free f b) -> Free f b
>>- a -> Free f b
f = a -> Free f b
f a
a
  Free f (Free f a)
m >>- a -> Free f b
f = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free ((forall (m :: * -> *) a b. Bind m => m a -> (a -> m b) -> m b
>>- a -> Free f b
f) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
m)

instance Functor f => Monad (Free f) where
  return :: forall a. a -> Free f a
return = forall (f :: * -> *) a. Applicative f => a -> f a
pure
  {-# INLINE return #-}
  Pure a
a >>= :: forall a b. Free f a -> (a -> Free f b) -> Free f b
>>= a -> Free f b
f = a -> Free f b
f a
a
  Free f (Free f a)
m >>= a -> Free f b
f = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free ((forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= a -> Free f b
f) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
m)

instance Functor f => MonadFix (Free f) where
  mfix :: forall a. (a -> Free f a) -> Free f a
mfix a -> Free f a
f = Free f a
a where a :: Free f a
a = a -> Free f a
f (forall {f :: * -> *} {a}. Free f a -> a
impure Free f a
a); impure :: Free f a -> a
impure (Pure a
x) = a
x; impure (Free f (Free f a)
_) = forall a. HasCallStack => String -> a
error String
"mfix (Free f): Free"

-- | This violates the Alternative laws, handle with care.

instance Alternative v => Alternative (Free v) where
  empty :: forall a. Free v a
empty = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall (f :: * -> *) a. Alternative f => f a
empty
  {-# INLINE empty #-}
  Free v a
a <|> :: forall a. Free v a -> Free v a -> Free v a
<|> Free v a
b = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (forall (f :: * -> *) a. Applicative f => a -> f a
pure Free v a
a forall (f :: * -> *) a. Alternative f => f a -> f a -> f a
<|> forall (f :: * -> *) a. Applicative f => a -> f a
pure Free v a
b)
  {-# INLINE (<|>) #-}

-- | This violates the MonadPlus laws, handle with care.

instance (Functor v, MonadPlus v) => MonadPlus (Free v) where
  mzero :: forall a. Free v a
mzero = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall (m :: * -> *) a. MonadPlus m => m a
mzero
  {-# INLINE mzero #-}
  Free v a
a mplus :: forall a. Free v a -> Free v a -> Free v a
`mplus` Free v a
b = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (forall (m :: * -> *) a. Monad m => a -> m a
return Free v a
a forall (m :: * -> *) a. MonadPlus m => m a -> m a -> m a
`mplus` forall (m :: * -> *) a. Monad m => a -> m a
return Free v a
b)
  {-# INLINE mplus #-}

-- | This is not a true monad transformer. It is only a monad transformer \"up to 'retract'\".

instance MonadTrans Free where
  lift :: forall (m :: * -> *) a. Monad m => m a -> Free m a
lift = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM forall (f :: * -> *) a. a -> Free f a
Pure
  {-# INLINE lift #-}

instance Foldable f => Foldable (Free f) where
  foldMap :: forall m a. Monoid m => (a -> m) -> Free f a -> m
foldMap a -> m
f = forall {t :: * -> *}. Foldable t => Free t a -> m
go where
    go :: Free t a -> m
go (Pure a
a) = a -> m
f a
a
    go (Free t (Free t a)
fa) = forall (t :: * -> *) m a.
(Foldable t, Monoid m) =>
(a -> m) -> t a -> m
foldMap Free t a -> m
go t (Free t a)
fa
  {-# INLINE foldMap #-}

  foldr :: forall a b. (a -> b -> b) -> b -> Free f a -> b
foldr a -> b -> b
f = forall {t :: * -> *}. Foldable t => b -> Free t a -> b
go where
    go :: b -> Free t a -> b
go b
r Free t a
free =
      case Free t a
free of
        Pure a
a -> a -> b -> b
f a
a b
r
        Free t (Free t a)
fa -> forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (forall a b c. (a -> b -> c) -> b -> a -> c
flip b -> Free t a -> b
go) b
r t (Free t a)
fa
  {-# INLINE foldr #-}

#if MIN_VERSION_base(4,6,0)
  foldl' :: forall b a. (b -> a -> b) -> b -> Free f a -> b
foldl' b -> a -> b
f = forall {t :: * -> *}. Foldable t => b -> Free t a -> b
go where
    go :: b -> Free t a -> b
go b
r Free t a
free =
      case Free t a
free of
        Pure a
a -> b -> a -> b
f b
r a
a
        Free t (Free t a)
fa -> forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
foldl' b -> Free t a -> b
go b
r t (Free t a)
fa
  {-# INLINE foldl' #-}
#endif

instance Foldable1 f => Foldable1 (Free f) where
  foldMap1 :: forall m a. Semigroup m => (a -> m) -> Free f a -> m
foldMap1 a -> m
f = forall {t :: * -> *}. Foldable1 t => Free t a -> m
go where
    go :: Free t a -> m
go (Pure a
a) = a -> m
f a
a
    go (Free t (Free t a)
fa) = forall (t :: * -> *) m a.
(Foldable1 t, Semigroup m) =>
(a -> m) -> t a -> m
foldMap1 Free t a -> m
go t (Free t a)
fa
  {-# INLINE foldMap1 #-}

instance Traversable f => Traversable (Free f) where
  traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Free f a -> f (Free f b)
traverse a -> f b
f = forall {f :: * -> *}. Traversable f => Free f a -> f (Free f b)
go where
    go :: Free f a -> f (Free f b)
go (Pure a
a) = forall (f :: * -> *) a. a -> Free f a
Pure forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> f b
f a
a
    go (Free f (Free f a)
fa) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
traverse Free f a -> f (Free f b)
go f (Free f a)
fa
  {-# INLINE traverse #-}

instance Traversable1 f => Traversable1 (Free f) where
  traverse1 :: forall (f :: * -> *) a b.
Apply f =>
(a -> f b) -> Free f a -> f (Free f b)
traverse1 a -> f b
f = forall {f :: * -> *}. Traversable1 f => Free f a -> f (Free f b)
go where
    go :: Free f a -> f (Free f b)
go (Pure a
a) = forall (f :: * -> *) a. a -> Free f a
Pure forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> f b
f a
a
    go (Free f (Free f a)
fa) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall (t :: * -> *) (f :: * -> *) a b.
(Traversable1 t, Apply f) =>
(a -> f b) -> t a -> f (t b)
traverse1 Free f a -> f (Free f b)
go f (Free f a)
fa
  {-# INLINE traverse1 #-}

instance FunctorWithIndex i f => FunctorWithIndex [i] (Free f) where
  imap :: forall a b. ([i] -> a -> b) -> Free f a -> Free f b
imap [i] -> a -> b
f (Pure a
a) = forall (f :: * -> *) a. a -> Free f a
Pure forall a b. (a -> b) -> a -> b
$ [i] -> a -> b
f [] a
a
  imap [i] -> a -> b
f (Free f (Free f a)
s) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ forall i (f :: * -> *) a b.
FunctorWithIndex i f =>
(i -> a -> b) -> f a -> f b
imap (\i
i -> forall i (f :: * -> *) a b.
FunctorWithIndex i f =>
(i -> a -> b) -> f a -> f b
imap ([i] -> a -> b
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. (:) i
i)) f (Free f a)
s
  {-# INLINE imap #-}

instance FoldableWithIndex i f => FoldableWithIndex [i] (Free f) where
  ifoldMap :: forall m a. Monoid m => ([i] -> a -> m) -> Free f a -> m
ifoldMap [i] -> a -> m
f (Pure a
a) = [i] -> a -> m
f [] a
a
  ifoldMap [i] -> a -> m
f (Free f (Free f a)
s) = forall i (f :: * -> *) m a.
(FoldableWithIndex i f, Monoid m) =>
(i -> a -> m) -> f a -> m
ifoldMap (\i
i -> forall i (f :: * -> *) m a.
(FoldableWithIndex i f, Monoid m) =>
(i -> a -> m) -> f a -> m
ifoldMap ([i] -> a -> m
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. (:) i
i)) f (Free f a)
s
  {-# INLINE ifoldMap #-}

instance TraversableWithIndex i f => TraversableWithIndex [i] (Free f) where
  itraverse :: forall (f :: * -> *) a b.
Applicative f =>
([i] -> a -> f b) -> Free f a -> f (Free f b)
itraverse [i] -> a -> f b
f (Pure a
a) = forall (f :: * -> *) a. a -> Free f a
Pure forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [i] -> a -> f b
f [] a
a
  itraverse [i] -> a -> f b
f (Free f (Free f a)
s) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall i (t :: * -> *) (f :: * -> *) a b.
(TraversableWithIndex i t, Applicative f) =>
(i -> a -> f b) -> t a -> f (t b)
itraverse (\i
i -> forall i (t :: * -> *) (f :: * -> *) a b.
(TraversableWithIndex i t, Applicative f) =>
(i -> a -> f b) -> t a -> f (t b)
itraverse ([i] -> a -> f b
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. (:) i
i)) f (Free f a)
s
  {-# INLINE itraverse #-}

instance (Functor m, MonadWriter e m) => MonadWriter e (Free m) where
  tell :: e -> Free m ()
tell = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall w (m :: * -> *). MonadWriter w m => w -> m ()
tell
  {-# INLINE tell #-}
  listen :: forall a. Free m a -> Free m (a, e)
listen = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall w (m :: * -> *) a. MonadWriter w m => m a -> m (a, w)
listen forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Monad f => Free f a -> f a
retract
  {-# INLINE listen #-}
  pass :: forall a. Free m (a, e -> e) -> Free m a
pass = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall w (m :: * -> *) a. MonadWriter w m => m (a, w -> w) -> m a
pass forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Monad f => Free f a -> f a
retract
  {-# INLINE pass #-}

instance (Functor m, MonadReader e m) => MonadReader e (Free m) where
  ask :: Free m e
ask = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall r (m :: * -> *). MonadReader r m => m r
ask
  {-# INLINE ask #-}
  local :: forall a. (e -> e) -> Free m a -> Free m a
local e -> e
f = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall r (m :: * -> *) a. MonadReader r m => (r -> r) -> m a -> m a
local e -> e
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Monad f => Free f a -> f a
retract
  {-# INLINE local #-}

instance (Functor m, MonadState s m) => MonadState s (Free m) where
  get :: Free m s
get = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall s (m :: * -> *). MonadState s m => m s
get
  {-# INLINE get #-}
  put :: s -> Free m ()
put s
s = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift (forall s (m :: * -> *). MonadState s m => s -> m ()
put s
s)
  {-# INLINE put #-}

instance (Functor m, MonadError e m) => MonadError e (Free m) where
  throwError :: forall a. e -> Free m a
throwError = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall e (m :: * -> *) a. MonadError e m => e -> m a
throwError
  {-# INLINE throwError #-}
  catchError :: forall a. Free m a -> (e -> Free m a) -> Free m a
catchError Free m a
as e -> Free m a
f = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift (forall e (m :: * -> *) a.
MonadError e m =>
m a -> (e -> m a) -> m a
catchError (forall (f :: * -> *) a. Monad f => Free f a -> f a
retract Free m a
as) (forall (f :: * -> *) a. Monad f => Free f a -> f a
retract forall b c a. (b -> c) -> (a -> b) -> a -> c
. e -> Free m a
f))
  {-# INLINE catchError #-}

instance (Functor m, MonadCont m) => MonadCont (Free m) where
  callCC :: forall a b. ((a -> Free m b) -> Free m a) -> Free m a
callCC (a -> Free m b) -> Free m a
f = forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift (forall (m :: * -> *) a b. MonadCont m => ((a -> m b) -> m a) -> m a
callCC (forall (f :: * -> *) a. Monad f => Free f a -> f a
retract forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Free m b) -> Free m a
f forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift))
  {-# INLINE callCC #-}

instance Functor f => MonadFree f (Free f) where
  wrap :: forall a. f (Free f a) -> Free f a
wrap = forall (f :: * -> *) a. f (Free f a) -> Free f a
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 :: forall (f :: * -> *) a. Monad f => Free f a -> f a
retract (Pure a
a) = forall (m :: * -> *) a. Monad m => a -> m a
return a
a
retract (Free f (Free f a)
as) = f (Free f a)
as forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= forall (f :: * -> *) a. Monad f => Free f a -> f a
retract

-- | Tear down a 'Free' 'Monad' using iteration.

iter :: Functor f => (f a -> a) -> Free f a -> a
iter :: forall (f :: * -> *) a. Functor f => (f a -> a) -> Free f a -> a
iter f a -> a
_ (Pure a
a) = a
a
iter f a -> a
phi (Free f (Free f a)
m) = f a -> a
phi (forall (f :: * -> *) a. Functor f => (f a -> a) -> Free f a -> a
iter f a -> a
phi forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
m)

-- | Like 'iter' for applicative values.

iterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a
iterA :: forall (p :: * -> *) (f :: * -> *) a.
(Applicative p, Functor f) =>
(f (p a) -> p a) -> Free f a -> p a
iterA f (p a) -> p a
_   (Pure a
x) = forall (f :: * -> *) a. Applicative f => a -> f a
pure a
x
iterA f (p a) -> p a
phi (Free f (Free f a)
f) = f (p a) -> p a
phi (forall (p :: * -> *) (f :: * -> *) a.
(Applicative p, Functor f) =>
(f (p a) -> p a) -> Free f a -> p a
iterA f (p a) -> p a
phi forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
f)

-- | Like 'iter' for monadic values.

iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a
iterM :: forall (m :: * -> *) (f :: * -> *) a.
(Monad m, Functor f) =>
(f (m a) -> m a) -> Free f a -> m a
iterM f (m a) -> m a
_   (Pure a
x) = forall (m :: * -> *) a. Monad m => a -> m a
return a
x
iterM f (m a) -> m a
phi (Free f (Free f a)
f) = f (m a) -> m a
phi (forall (m :: * -> *) (f :: * -> *) a.
(Monad m, Functor f) =>
(f (m a) -> m a) -> Free f a -> m a
iterM f (m a) -> m a
phi forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f a)
f)

-- | Lift a natural transformation from @f@ to @g@ into a natural transformation from @'Free' f@ to @'Free' g@.

hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b
hoistFree :: forall (g :: * -> *) (f :: * -> *) b.
Functor g =>
(forall a. f a -> g a) -> Free f b -> Free g b
hoistFree forall a. f a -> g a
_ (Pure b
a)  = forall (f :: * -> *) a. a -> Free f a
Pure b
a
hoistFree forall a. f a -> g a
f (Free f (Free f b)
as) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (forall (g :: * -> *) (f :: * -> *) b.
Functor g =>
(forall a. f a -> g a) -> Free f b -> Free g b
hoistFree forall a. f a -> g a
f forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. f a -> g a
f f (Free f b)
as)

-- | The very definition of a free monad is that given a natural transformation you get a monad homomorphism.

foldFree :: Monad m => (forall x . f x -> m x) -> Free f a -> m a
foldFree :: forall (m :: * -> *) (f :: * -> *) a.
Monad m =>
(forall x. f x -> m x) -> Free f a -> m a
foldFree forall x. f x -> m x
_ (Pure a
a)  = forall (m :: * -> *) a. Monad m => a -> m a
return a
a
foldFree forall x. f x -> m x
f (Free f (Free f a)
as) = forall x. f x -> m x
f f (Free f a)
as forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= forall (m :: * -> *) (f :: * -> *) a.
Monad m =>
(forall x. f x -> m x) -> Free f a -> m a
foldFree forall x. f x -> m x
f

-- | Convert a 'Free' monad from "Control.Monad.Free" to a 'FreeT.FreeT' monad

-- from "Control.Monad.Trans.Free".

toFreeT :: (Functor f, Monad m) => Free f a -> FreeT.FreeT f m a
toFreeT :: forall (f :: * -> *) (m :: * -> *) a.
(Functor f, Monad m) =>
Free f a -> FreeT f m a
toFreeT (Pure a
a) = forall (f :: * -> *) (m :: * -> *) a.
m (FreeF f a (FreeT f m a)) -> FreeT f m a
FreeT.FreeT (forall (m :: * -> *) a. Monad m => a -> m a
return (forall (f :: * -> *) a b. a -> FreeF f a b
FreeT.Pure a
a))
toFreeT (Free f (Free f a)
f) = forall (f :: * -> *) (m :: * -> *) a.
m (FreeF f a (FreeT f m a)) -> FreeT f m a
FreeT.FreeT (forall (m :: * -> *) a. Monad m => a -> m a
return (forall (f :: * -> *) a b. f b -> FreeF f a b
FreeT.Free (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall (f :: * -> *) (m :: * -> *) a.
(Functor f, Monad m) =>
Free f a -> FreeT f m a
toFreeT f (Free f a)
f)))

-- | 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 :: (Functor f) => Integer -> Free f a -> Free f (Maybe a)
cutoff :: forall (f :: * -> *) a.
Functor f =>
Integer -> Free f a -> Free f (Maybe a)
cutoff Integer
n Free f a
_ | Integer
n forall a. Ord a => a -> a -> Bool
<= Integer
0 = forall (m :: * -> *) a. Monad m => a -> m a
return forall a. Maybe a
Nothing
cutoff Integer
n (Free f (Free f a)
f) = forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall a b. (a -> b) -> a -> b
$ forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (forall (f :: * -> *) a.
Functor f =>
Integer -> Free f a -> Free f (Maybe a)
cutoff (Integer
n forall a. Num a => a -> a -> a
- Integer
1)) f (Free f a)
f
cutoff Integer
_ Free f a
m = forall a. a -> Maybe a
Just forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Free f a
m

-- | Unfold a free monad from a seed.

unfold :: Functor f => (b -> Either a (f b)) -> b -> Free f a
unfold :: forall (f :: * -> *) b a.
Functor f =>
(b -> Either a (f b)) -> b -> Free f a
unfold b -> Either a (f b)
f = b -> Either a (f b)
f forall {k} (cat :: k -> k -> *) (a :: k) (b :: k) (c :: k).
Category cat =>
cat a b -> cat b c -> cat a c
>>> forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either forall (f :: * -> *) a. a -> Free f a
Pure (forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (forall (f :: * -> *) b a.
Functor f =>
(b -> Either a (f b)) -> b -> Free f a
unfold b -> Either a (f b)
f))

-- | Unfold a free monad from a seed, monadically.

unfoldM :: (Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a)
unfoldM :: forall (f :: * -> *) (m :: * -> *) b a.
(Traversable f, Applicative m, Monad m) =>
(b -> m (Either a (f b))) -> b -> m (Free f a)
unfoldM b -> m (Either a (f b))
f = b -> m (Either a (f b))
f forall (m :: * -> *) a b c.
Monad m =>
(a -> m b) -> (b -> m c) -> a -> m c
>=> forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (forall (f :: * -> *) a. Applicative f => a -> f a
pure forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: * -> *) a. Applicative f => a -> f a
pure) (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall (f :: * -> *) a. f (Free f a) -> Free f a
Free forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
traverse (forall (f :: * -> *) (m :: * -> *) b a.
(Traversable f, Applicative m, Monad m) =>
(b -> m (Either a (f b))) -> b -> m (Free f a)
unfoldM b -> m (Either a (f b))
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 :: forall (f :: * -> *) (m :: * -> *) a (p :: * -> * -> *).
(Choice p, Applicative m) =>
p a (m a) -> p (Free f a) (m (Free f a))
_Pure = forall (p :: * -> * -> *) a b c d.
Profunctor p =>
(a -> b) -> (c -> d) -> p b c -> p a d
dimap forall {f :: * -> *} {b}. Free f b -> Either (Free f b) b
impure (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either forall (f :: * -> *) a. Applicative f => a -> f a
pure (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall (f :: * -> *) a. a -> Free f a
Pure)) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (p :: * -> * -> *) a b c.
Choice p =>
p a b -> p (Either c a) (Either c b)
right'
 where
  impure :: Free f b -> Either (Free f b) b
impure (Pure b
x) = forall a b. b -> Either a b
Right b
x
  impure Free f b
x        = forall a b. a -> Either a b
Left Free f b
x
  {-# INLINE impure #-}
{-# INLINE _Pure #-}

-- | This is @Prism (Free f a) (Free g a) (f (Free f a)) (g (Free g 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 g m a p. (Choice p, Applicative m)
      => p (f (Free f a)) (m (g (Free g a))) -> p (Free f a) (m (Free g a))
_Free :: forall (f :: * -> *) (g :: * -> *) (m :: * -> *) a
       (p :: * -> * -> *).
(Choice p, Applicative m) =>
p (f (Free f a)) (m (g (Free g a))) -> p (Free f a) (m (Free g a))
_Free = forall (p :: * -> * -> *) a b c d.
Profunctor p =>
(a -> b) -> (c -> d) -> p b c -> p a d
dimap forall {f :: * -> *} {a} {f :: * -> *}.
Free f a -> Either (Free f a) (f (Free f a))
unfree (forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either forall (f :: * -> *) a. Applicative f => a -> f a
pure (forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap forall (f :: * -> *) a. f (Free f a) -> Free f a
Free)) forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (p :: * -> * -> *) a b c.
Choice p =>
p a b -> p (Either c a) (Either c b)
right'
 where
  unfree :: Free f a -> Either (Free f a) (f (Free f a))
unfree (Free f (Free f a)
x) = forall a b. b -> Either a b
Right f (Free f a)
x
  unfree (Pure a
x) = forall a b. a -> Either a b
Left (forall (f :: * -> *) a. a -> Free f a
Pure a
x)
  {-# INLINE unfree #-}
{-# INLINE _Free #-}


#if __GLASGOW_HASKELL__ < 707
instance Typeable1 f => Typeable1 (Free f) where
  typeOf1 t = mkTyConApp freeTyCon [typeOf1 (f t)] where
    f :: Free f a -> f a
    f = undefined

freeTyCon :: TyCon
#if __GLASGOW_HASKELL__ < 704
freeTyCon = mkTyCon "Control.Monad.Free.Free"
#else
freeTyCon = mkTyCon3 "free" "Control.Monad.Free" "Free"
#endif
{-# NOINLINE freeTyCon #-}

instance
  ( Typeable1 f, Typeable a
  , Data a, Data (f (Free f a))
  ) => Data (Free f a) where
    gfoldl f z (Pure a) = z Pure `f` a
    gfoldl f z (Free as) = z Free `f` as
    toConstr Pure{} = pureConstr
    toConstr Free{} = freeConstr
    gunfold k z c = case constrIndex c of
        1 -> k (z Pure)
        2 -> k (z Free)
        _ -> error "gunfold"
    dataTypeOf _ = freeDataType
    dataCast1 f = gcast1 f

pureConstr, freeConstr :: Constr
pureConstr = mkConstr freeDataType "Pure" [] Prefix
freeConstr = mkConstr freeDataType "Free" [] Prefix
{-# NOINLINE pureConstr #-}
{-# NOINLINE freeConstr #-}

freeDataType :: DataType
freeDataType = mkDataType "Control.Monad.Free.FreeF" [pureConstr, freeConstr]
{-# NOINLINE freeDataType #-}

#endif