{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE Safe #-}

-----------------------------------------------------------------------------
-- |
-- 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
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 GHC.Generics
import Prelude hiding (foldr)

-- $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))
  deriving ((forall x. Free f a -> Rep (Free f a) x)
-> (forall x. Rep (Free f a) x -> Free f a) -> Generic (Free f a)
forall x. Rep (Free f a) x -> Free f a
forall x. Free f a -> Rep (Free f a) x
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
$cfrom :: forall (f :: * -> *) a x. Free f a -> Rep (Free f a) x
from :: forall x. Free f a -> Rep (Free f a) x
$cto :: forall (f :: * -> *) a x. Rep (Free f a) x -> Free f a
to :: forall x. Rep (Free f a) x -> Free f a
Generic, (forall a. Free f a -> Rep1 (Free f) a)
-> (forall a. Rep1 (Free f) a -> Free f a) -> Generic1 (Free f)
forall a. Rep1 (Free f) a -> Free f a
forall a. Free f a -> Rep1 (Free f) a
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
$cfrom1 :: forall (f :: * -> *) a. Functor f => Free f a -> Rep1 (Free f) a
from1 :: forall a. Free f a -> Rep1 (Free f) a
$cto1 :: forall (f :: * -> *) a. Functor f => Rep1 (Free f) a -> Free f a
to1 :: forall a. Rep1 (Free f) a -> Free f a
Generic1)

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

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 = Free f a -> Free f b -> Bool
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) = (Free f a -> Free f b -> Bool)
-> f (Free f a) -> f (Free f b) -> Bool
forall a b. (a -> b -> Bool) -> f a -> f b -> Bool
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

instance (Eq1 f, Eq a) => Eq (Free f a) where
  == :: Free f a -> Free f a -> Bool
(==) = Free f a -> Free f a -> Bool
forall (f :: * -> *) a. (Eq1 f, Eq a) => f a -> f a -> Bool
eq1

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 = Free f a -> Free f b -> Ordering
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) = (Free f a -> Free f b -> Ordering)
-> f (Free f a) -> f (Free f b) -> Ordering
forall a b. (a -> b -> Ordering) -> f a -> f b -> Ordering
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

instance (Ord1 f, Ord a) => Ord (Free f a) where
  compare :: Free f a -> Free f a -> Ordering
compare = Free f a -> Free f a -> Ordering
forall (f :: * -> *) a. (Ord1 f, Ord a) => f a -> f a -> Ordering
compare1

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 = Int -> Free f a -> ShowS
forall {f :: * -> *}. Show1 f => Int -> Free f a -> ShowS
go
    where
      go :: Int -> Free f a -> ShowS
go Int
d (Pure a
a) = (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
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) = (Int -> f (Free f a) -> ShowS)
-> String -> Int -> f (Free f a) -> ShowS
forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith ((Int -> Free f a -> ShowS)
-> ([Free f a] -> ShowS) -> Int -> f (Free f a) -> ShowS
forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
liftShowsPrec Int -> Free f a -> ShowS
go ((Int -> a -> ShowS) -> ([a] -> ShowS) -> [Free f a] -> ShowS
forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> [Free f a] -> ShowS
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

instance (Show1 f, Show a) => Show (Free f a) where
  showsPrec :: Int -> Free f a -> ShowS
showsPrec = Int -> Free f a -> ShowS
forall (f :: * -> *) a. (Show1 f, Show a) => Int -> f a -> ShowS
showsPrec1

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 = (String -> ReadS (Free f a)) -> Int -> ReadS (Free f a)
forall a. (String -> ReadS a) -> Int -> ReadS a
readsData ((String -> ReadS (Free f a)) -> Int -> ReadS (Free f a))
-> (String -> ReadS (Free f a)) -> Int -> ReadS (Free f a)
forall a b. (a -> b) -> a -> b
$
        (Int -> ReadS a)
-> String -> (a -> Free f a) -> String -> ReadS (Free f a)
forall a t.
(Int -> ReadS a) -> String -> (a -> t) -> String -> ReadS t
readsUnaryWith Int -> ReadS a
rp String
"Pure" a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure (String -> ReadS (Free f a))
-> (String -> ReadS (Free f a)) -> String -> ReadS (Free f a)
forall a. Monoid a => a -> a -> a
`mappend`
        (Int -> ReadS (f (Free f a)))
-> String
-> (f (Free f a) -> Free f a)
-> String
-> ReadS (Free f a)
forall a t.
(Int -> ReadS a) -> String -> (a -> t) -> String -> ReadS t
readsUnaryWith ((Int -> ReadS (Free f a))
-> ReadS [Free f a] -> Int -> ReadS (f (Free f a))
forall a. (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (f a)
forall (f :: * -> *) a.
Read1 f =>
(Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (f a)
liftReadsPrec Int -> ReadS (Free f a)
go ((Int -> ReadS a) -> ReadS [a] -> ReadS [Free f a]
forall a. (Int -> ReadS a) -> ReadS [a] -> ReadS [Free f a]
forall (f :: * -> *) a.
Read1 f =>
(Int -> ReadS a) -> ReadS [a] -> ReadS [f a]
liftReadList Int -> ReadS a
rp ReadS [a]
rl)) String
"Free" f (Free f a) -> Free f a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free

instance (Read1 f, Read a) => Read (Free f a) where
  readsPrec :: Int -> ReadS (Free f a)
readsPrec = Int -> ReadS (Free f a)
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 = Free f a -> Free f b
forall {f :: * -> *}. Functor f => Free f a -> Free f b
go where
    go :: Free f a -> Free f b
go (Pure a
a)  = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (a -> b
f a
a)
    go (Free f (Free f a)
fa) = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (Free f a -> Free f b
go (Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
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 = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (a -> b
a a
b)
  Pure a -> b
a  <.> Free f (Free f a)
fb = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (Free f b) -> Free f b
forall a b. (a -> b) -> a -> b
$ (a -> b) -> Free f a -> Free f b
forall a b. (a -> b) -> Free f a -> Free f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
a (Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
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 = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (Free f b) -> Free f b
forall a b. (a -> b) -> a -> b
$ (Free f (a -> b) -> Free f a -> Free f b
forall a b. Free f (a -> b) -> Free f a -> Free f b
forall (f :: * -> *) a b. Apply f => f (a -> b) -> f a -> f b
<.> Free f a
b) (Free f (a -> b) -> Free f b)
-> f (Free f (a -> b)) -> f (Free f 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 = a -> Free f a
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 = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (b -> Free f b) -> b -> Free f b
forall a b. (a -> b) -> a -> b
$ a -> b
a a
b
  Pure a -> b
a <*> Free f (Free f a)
mb = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (Free f b) -> Free f b
forall a b. (a -> b) -> a -> b
$ (a -> b) -> Free f a -> Free f b
forall a b. (a -> b) -> Free f a -> Free f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
a (Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
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 = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (Free f b) -> Free f b
forall a b. (a -> b) -> a -> b
$ (Free f (a -> b) -> Free f a -> Free f b
forall a b. Free f (a -> b) -> Free f a -> Free f b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> Free f a
b) (Free f (a -> b) -> Free f b)
-> f (Free f (a -> b)) -> f (Free f 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 = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free ((Free f a -> (a -> Free f b) -> Free f b
forall a b. Free f a -> (a -> Free f b) -> Free f b
forall (m :: * -> *) a b. Bind m => m a -> (a -> m b) -> m b
>>- a -> Free f b
f) (Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
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 = a -> Free f a
forall a. a -> Free f a
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 = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free ((Free f a -> (a -> Free f b) -> Free f b
forall a b. Free f a -> (a -> Free f b) -> Free f b
forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= a -> Free f b
f) (Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
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 (Free f a -> a
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)
_) = String -> 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 = v (Free v a) -> Free v a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free v (Free v a)
forall a. v a
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 = v (Free v a) -> Free v a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (Free v a -> v (Free v a)
forall a. a -> v a
forall (f :: * -> *) a. Applicative f => a -> f a
pure Free v a
a v (Free v a) -> v (Free v a) -> v (Free v a)
forall a. v a -> v a -> v a
forall (f :: * -> *) a. Alternative f => f a -> f a -> f a
<|> Free v a -> v (Free v a)
forall a. a -> v a
forall (f :: * -> *) a. Applicative f => a -> f a
pure Free v a
b)
  {-# INLINE (<|>) #-}

-- | This violates the MonadPlus laws, handle with care.
instance MonadPlus v => MonadPlus (Free v) where
  mzero :: forall a. Free v a
mzero = v (Free v a) -> Free v a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free v (Free v a)
forall a. v a
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 = v (Free v a) -> Free v a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (Free v a -> v (Free v a)
forall a. a -> v a
forall (m :: * -> *) a. Monad m => a -> m a
return Free v a
a v (Free v a) -> v (Free v a) -> v (Free v a)
forall a. v a -> v a -> v a
forall (m :: * -> *) a. MonadPlus m => m a -> m a -> m a
`mplus` Free v a -> v (Free v a)
forall a. a -> v a
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 = m (Free m a) -> Free m a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (m (Free m a) -> Free m a)
-> (m a -> m (Free m a)) -> m a -> Free m a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Free m a) -> m a -> m (Free m a)
forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM a -> Free m a
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 = Free f a -> m
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) = (Free t a -> m) -> t (Free t a) -> m
forall m a. Monoid m => (a -> m) -> t a -> m
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 = b -> Free f a -> b
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 -> (Free t a -> b -> b) -> b -> t (Free t a) -> b
forall a b. (a -> b -> b) -> b -> t a -> b
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr ((b -> Free t a -> b) -> Free t a -> b -> b
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 #-}

  foldl' :: forall b a. (b -> a -> b) -> b -> Free f a -> b
foldl' b -> a -> b
f = b -> Free f a -> b
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 -> (b -> Free t a -> b) -> b -> t (Free t a) -> b
forall b a. (b -> a -> b) -> b -> t a -> b
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' #-}

instance Foldable1 f => Foldable1 (Free f) where
  foldMap1 :: forall m a. Semigroup m => (a -> m) -> Free f a -> m
foldMap1 a -> m
f = Free f a -> m
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) = (Free t a -> m) -> t (Free t a) -> m
forall m a. Semigroup m => (a -> m) -> t a -> m
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 = Free f a -> f (Free f b)
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) = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (b -> Free f b) -> f b -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> f b
f a
a
    go (Free f (Free f a)
fa) = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (f (Free f b)) -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (Free f a -> f (Free f b)) -> f (Free f a) -> f (f (Free f b))
forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> f a -> f (f 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 = Free f a -> f (Free f b)
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) = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (b -> Free f b) -> f b -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> f b
f a
a
    go (Free f (Free f a)
fa) = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (f (Free f b)) -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (Free f a -> f (Free f b)) -> f (Free f a) -> f (f (Free f b))
forall (t :: * -> *) (f :: * -> *) a b.
(Traversable1 t, Apply f) =>
(a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b. Apply f => (a -> f b) -> f a -> f (f 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) = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (b -> Free f b) -> b -> Free f b
forall a b. (a -> b) -> a -> b
$ [i] -> a -> b
f [] a
a
  imap [i] -> a -> b
f (Free f (Free f a)
s) = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (Free f b) -> Free f b
forall a b. (a -> b) -> a -> b
$ (i -> Free f a -> Free f b) -> f (Free f a) -> f (Free f b)
forall a b. (i -> a -> b) -> f a -> f b
forall i (f :: * -> *) a b.
FunctorWithIndex i f =>
(i -> a -> b) -> f a -> f b
imap (\i
i -> ([i] -> a -> b) -> Free f a -> Free f b
forall a b. ([i] -> a -> b) -> Free f a -> Free f b
forall i (f :: * -> *) a b.
FunctorWithIndex i f =>
(i -> a -> b) -> f a -> f b
imap ([i] -> a -> b
f ([i] -> a -> b) -> ([i] -> [i]) -> [i] -> a -> b
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) = (i -> Free f a -> m) -> f (Free f a) -> m
forall m a. Monoid m => (i -> a -> m) -> f a -> m
forall i (f :: * -> *) m a.
(FoldableWithIndex i f, Monoid m) =>
(i -> a -> m) -> f a -> m
ifoldMap (\i
i -> ([i] -> a -> m) -> Free f a -> m
forall m a. Monoid m => ([i] -> a -> m) -> Free f a -> m
forall i (f :: * -> *) m a.
(FoldableWithIndex i f, Monoid m) =>
(i -> a -> m) -> f a -> m
ifoldMap ([i] -> a -> m
f ([i] -> a -> m) -> ([i] -> [i]) -> [i] -> a -> m
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) = b -> Free f b
forall (f :: * -> *) a. a -> Free f a
Pure (b -> Free f b) -> f b -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [i] -> a -> f b
f [] a
a
  itraverse [i] -> a -> f b
f (Free f (Free f a)
s) = f (Free f b) -> Free f b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f b) -> Free f b) -> f (f (Free f b)) -> f (Free f b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (i -> Free f a -> f (Free f b)) -> f (Free f a) -> f (f (Free f b))
forall i (t :: * -> *) (f :: * -> *) a b.
(TraversableWithIndex i t, Applicative f) =>
(i -> a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
(i -> a -> f b) -> f a -> f (f b)
itraverse (\i
i -> ([i] -> a -> f b) -> Free f a -> f (Free f b)
forall i (t :: * -> *) (f :: * -> *) a b.
(TraversableWithIndex i t, Applicative f) =>
(i -> a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
([i] -> a -> f b) -> Free f a -> f (Free f b)
itraverse ([i] -> a -> f b
f ([i] -> a -> f b) -> ([i] -> [i]) -> [i] -> a -> f b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (:) i
i)) f (Free f a)
s
  {-# INLINE itraverse #-}

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

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

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

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

instance 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 = m a -> Free m a
forall (m :: * -> *) a. Monad m => m a -> Free m a
forall (t :: (* -> *) -> * -> *) (m :: * -> *) a.
(MonadTrans t, Monad m) =>
m a -> t m a
lift (((a -> m b) -> m a) -> m a
forall a b. ((a -> m b) -> m a) -> m a
forall (m :: * -> *) a b. MonadCont m => ((a -> m b) -> m a) -> m a
callCC (Free m a -> m a
forall (f :: * -> *) a. Monad f => Free f a -> f a
retract (Free m a -> m a) -> ((a -> m b) -> Free m a) -> (a -> m b) -> m a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Free m b) -> Free m a
f ((a -> Free m b) -> Free m a)
-> ((a -> m b) -> a -> Free m b) -> (a -> m b) -> Free m a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (m b -> Free m b) -> (a -> m b) -> a -> Free m b
forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM m b -> Free m b
forall (m :: * -> *) a. Monad m => m a -> Free m a
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 = f (Free f a) -> Free f a
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) = a -> f a
forall a. a -> f a
forall (m :: * -> *) a. Monad m => a -> m a
return a
a
retract (Free f (Free f a)
as) = f (Free f a)
as f (Free f a) -> (Free f a -> f a) -> f a
forall a b. f a -> (a -> f b) -> f b
forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= Free f a -> f a
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 ((f a -> a) -> Free f a -> a
forall (f :: * -> *) a. Functor f => (f a -> a) -> Free f a -> a
iter f a -> a
phi (Free f a -> a) -> f (Free f a) -> f a
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) = a -> p a
forall a. a -> p a
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 ((f (p a) -> p a) -> Free f a -> p a
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 (Free f a -> p a) -> f (Free f a) -> f (p a)
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) = a -> m a
forall a. a -> m a
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 ((f (m a) -> m a) -> Free f a -> m a
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 (Free f a -> m a) -> f (Free f a) -> f (m a)
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)  = b -> Free g b
forall (f :: * -> *) a. a -> Free f a
Pure b
a
hoistFree forall a. f a -> g a
f (Free f (Free f b)
as) = g (Free g b) -> Free g b
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free ((forall a. f a -> g a) -> Free f b -> Free g b
forall (g :: * -> *) (f :: * -> *) b.
Functor g =>
(forall a. f a -> g a) -> Free f b -> Free g b
hoistFree f a -> g a
forall a. f a -> g a
f (Free f b -> Free g b) -> g (Free f b) -> g (Free g b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f b) -> g (Free 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)  = a -> m a
forall a. a -> m 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) = f (Free f a) -> m (Free f a)
forall x. f x -> m x
f f (Free f a)
as m (Free f a) -> (Free f a -> m a) -> m a
forall a b. m a -> (a -> m b) -> m b
forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= (forall x. f x -> m x) -> Free f a -> m a
forall (m :: * -> *) (f :: * -> *) a.
Monad m =>
(forall x. f x -> m x) -> Free f a -> m a
foldFree f x -> m x
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) = m (FreeF f a (FreeT f m a)) -> FreeT f m a
forall (f :: * -> *) (m :: * -> *) a.
m (FreeF f a (FreeT f m a)) -> FreeT f m a
FreeT.FreeT (FreeF f a (FreeT f m a) -> m (FreeF f a (FreeT f m a))
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (a -> FreeF f a (FreeT f m a)
forall (f :: * -> *) a b. a -> FreeF f a b
FreeT.Pure a
a))
toFreeT (Free f (Free f a)
f) = m (FreeF f a (FreeT f m a)) -> FreeT f m a
forall (f :: * -> *) (m :: * -> *) a.
m (FreeF f a (FreeT f m a)) -> FreeT f m a
FreeT.FreeT (FreeF f a (FreeT f m a) -> m (FreeF f a (FreeT f m a))
forall a. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return (f (FreeT f m a) -> FreeF f a (FreeT f m a)
forall (f :: * -> *) a b. f b -> FreeF f a b
FreeT.Free ((Free f a -> FreeT f m a) -> f (Free f a) -> f (FreeT f m a)
forall a b. (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Free f a -> FreeT f m a
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 Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer
0 = Maybe a -> Free f (Maybe a)
forall a. a -> Free f a
forall (m :: * -> *) a. Monad m => a -> m a
return Maybe a
forall a. Maybe a
Nothing
cutoff Integer
n (Free f (Free f a)
f) = f (Free f (Maybe a)) -> Free f (Maybe a)
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f (Maybe a)) -> Free f (Maybe a))
-> f (Free f (Maybe a)) -> Free f (Maybe a)
forall a b. (a -> b) -> a -> b
$ (Free f a -> Free f (Maybe a))
-> f (Free f a) -> f (Free f (Maybe a))
forall a b. (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (Integer -> Free f a -> Free f (Maybe a)
forall (f :: * -> *) a.
Functor f =>
Integer -> Free f a -> Free f (Maybe a)
cutoff (Integer
n Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
1)) f (Free f a)
f
cutoff Integer
_ Free f a
m = a -> Maybe a
forall a. a -> Maybe a
Just (a -> Maybe a) -> Free f a -> Free f (Maybe a)
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 (b -> Either a (f b))
-> (Either a (f b) -> Free f a) -> b -> Free f a
forall {k} (cat :: k -> k -> *) (a :: k) (b :: k) (c :: k).
Category cat =>
cat a b -> cat b c -> cat a c
>>> (a -> Free f a) -> (f b -> Free f a) -> Either a (f b) -> Free f a
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure (f (Free f a) -> Free f a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f a) -> Free f a)
-> (f b -> f (Free f a)) -> f b -> Free f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (b -> Free f a) -> f b -> f (Free f a)
forall a b. (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((b -> Either a (f b)) -> b -> Free f a
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, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a)
unfoldM :: forall (f :: * -> *) (m :: * -> *) b a.
(Traversable f, 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 (b -> m (Either a (f b)))
-> (Either a (f b) -> m (Free f a)) -> b -> m (Free f a)
forall (m :: * -> *) a b c.
Monad m =>
(a -> m b) -> (b -> m c) -> a -> m c
>=> (a -> m (Free f a))
-> (f b -> m (Free f a)) -> Either a (f b) -> m (Free f a)
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either (Free f a -> m (Free f a)
forall a. a -> m a
forall (f :: * -> *) a. Applicative f => a -> f a
pure (Free f a -> m (Free f a)) -> (a -> Free f a) -> a -> m (Free f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Free f a
forall a. a -> Free f a
forall (f :: * -> *) a. Applicative f => a -> f a
pure) ((f (Free f a) -> Free f a) -> m (f (Free f a)) -> m (Free f a)
forall a b. (a -> b) -> m a -> m b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap f (Free f a) -> Free f a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (m (f (Free f a)) -> m (Free f a))
-> (f b -> m (f (Free f a))) -> f b -> m (Free f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (b -> m (Free f a)) -> f b -> m (f (Free f a))
forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> f a -> f (f b)
traverse ((b -> m (Either a (f b))) -> b -> m (Free f a)
forall (f :: * -> *) (m :: * -> *) b a.
(Traversable f, 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 = (Free f a -> Either (Free f a) a)
-> (Either (Free f a) (m a) -> m (Free f a))
-> p (Either (Free f a) a) (Either (Free f a) (m a))
-> p (Free f a) (m (Free f a))
forall a b c d. (a -> b) -> (c -> d) -> p b c -> p a d
forall (p :: * -> * -> *) a b c d.
Profunctor p =>
(a -> b) -> (c -> d) -> p b c -> p a d
dimap Free f a -> Either (Free f a) a
forall {f :: * -> *} {b}. Free f b -> Either (Free f b) b
impure ((Free f a -> m (Free f a))
-> (m a -> m (Free f a)) -> Either (Free f a) (m a) -> m (Free f a)
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either Free f a -> m (Free f a)
forall a. a -> m a
forall (f :: * -> *) a. Applicative f => a -> f a
pure ((a -> Free f a) -> m a -> m (Free f a)
forall a b. (a -> b) -> m a -> m b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure)) (p (Either (Free f a) a) (Either (Free f a) (m a))
 -> p (Free f a) (m (Free f a)))
-> (p a (m a) -> p (Either (Free f a) a) (Either (Free f a) (m a)))
-> p a (m a)
-> p (Free f a) (m (Free f a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. p a (m a) -> p (Either (Free f a) a) (Either (Free f a) (m a))
forall a b c. p a b -> p (Either c a) (Either c b)
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) = b -> Either (Free f b) b
forall a b. b -> Either a b
Right b
x
  impure Free f b
x        = Free f b -> Either (Free f b) b
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 = (Free f a -> Either (Free g a) (f (Free f a)))
-> (Either (Free g a) (m (g (Free g a))) -> m (Free g a))
-> p (Either (Free g a) (f (Free f a)))
     (Either (Free g a) (m (g (Free g a))))
-> p (Free f a) (m (Free g a))
forall a b c d. (a -> b) -> (c -> d) -> p b c -> p a d
forall (p :: * -> * -> *) a b c d.
Profunctor p =>
(a -> b) -> (c -> d) -> p b c -> p a d
dimap Free f a -> Either (Free g a) (f (Free f a))
forall {f :: * -> *} {a} {f :: * -> *}.
Free f a -> Either (Free f a) (f (Free f a))
unfree ((Free g a -> m (Free g a))
-> (m (g (Free g a)) -> m (Free g a))
-> Either (Free g a) (m (g (Free g a)))
-> m (Free g a)
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either Free g a -> m (Free g a)
forall a. a -> m a
forall (f :: * -> *) a. Applicative f => a -> f a
pure ((g (Free g a) -> Free g a) -> m (g (Free g a)) -> m (Free g a)
forall a b. (a -> b) -> m a -> m b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap g (Free g a) -> Free g a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free)) (p (Either (Free g a) (f (Free f a)))
   (Either (Free g a) (m (g (Free g a))))
 -> p (Free f a) (m (Free g a)))
-> (p (f (Free f a)) (m (g (Free g a)))
    -> p (Either (Free g a) (f (Free f a)))
         (Either (Free g a) (m (g (Free g a)))))
-> p (f (Free f a)) (m (g (Free g a)))
-> p (Free f a) (m (Free g a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. p (f (Free f a)) (m (g (Free g a)))
-> p (Either (Free g a) (f (Free f a)))
     (Either (Free g a) (m (g (Free g a))))
forall a b c. p a b -> p (Either c a) (Either c b)
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) = f (Free f a) -> Either (Free f a) (f (Free f a))
forall a b. b -> Either a b
Right f (Free f a)
x
  unfree (Pure a
x) = Free f a -> Either (Free f a) (f (Free f a))
forall a b. a -> Either a b
Left (a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure a
x)
  {-# INLINE unfree #-}
{-# INLINE _Free #-}