{-# LANGUAGE CPP, TypeFamilies, Rank2Types, FlexibleContexts, FlexibleInstances, GADTs, StandaloneDeriving, UndecidableInstances #-}
#include "recursion-schemes-common.h"

#ifdef __GLASGOW_HASKELL__
{-# LANGUAGE DeriveDataTypeable #-}
#if __GLASGOW_HASKELL__ >= 800
{-# LANGUAGE ConstrainedClassMethods #-}
#endif
#if HAS_GENERIC
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE ScopedTypeVariables, DefaultSignatures, MultiParamTypeClasses, TypeOperators #-}
#endif
#endif

-----------------------------------------------------------------------------
-- |
-- Copyright   :  (C) 2008-2015 Edward Kmett
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  : "Samuel Gélineau" <gelisam@gmail.com>,
--               "Luc Tielen" <luc.tielen@gmail.com>,
--               "Ryan Scott" <ryan.gl.scott@gmail.com>
-- Stability   :  experimental
-- Portability :  non-portable
--
----------------------------------------------------------------------------
module Data.Functor.Foldable
  (
  -- * Base functors
    Base
  , ListF(..)
  -- * Type classes
  , Recursive(project)
  , Corecursive(embed)
  -- * Folding functions
  -- $foldingFunctions
  , fold
  , cata
  , cataA
  , para
  , histo
  , zygo
  -- * Unfolding functions
  , unfold
  , ana
  , apo
  , futu
  -- * Combining unfolds and folds
  , refold
  , hylo
  , chrono
  -- * Changing representation
  , refix
  , hoist
  , transverse
  , cotransverse
  -- * Advanced usage
  -- ** Mendler-style recursion-schemes
  , mcata
  , mpara
  , mhisto
  , mzygo
  , mana
  , mapo
  , mfutu
  -- ** Fokkinga's recursion-schemes
  , prepro
  , postpro
  -- ** Elgot (co)algebras
  , elgot
  , coelgot
  -- ** Generalized recursion-schemes
  , gfold
  , gcata
  , gpara
  , ghisto
  , gzygo
  , gunfold
  , gana
  , gapo
  , gfutu
  , grefold
  , ghylo
  , gchrono
  , gprepro
  , gpostpro
  , distCata
  , distPara
  , distParaT
  , distHisto
  , distGHisto
  , distZygo
  , distZygoT
  , distAna
  , distApo
  , distGApo
  , distGApoT
  , distFutu
  , distGFutu
  -- ** Zygohistomorphic prepromorphisms
  , zygoHistoPrepro
  ) where

import Control.Applicative
import Control.Comonad
import Control.Comonad.Trans.Class
import Control.Comonad.Trans.Env (EnvT(..))
import qualified Control.Comonad.Cofree as Cofree
import Control.Comonad.Cofree (Cofree(..))
import           Control.Comonad.Trans.Cofree (CofreeF, CofreeT(..))
import qualified Control.Comonad.Trans.Cofree as CCTC
import Control.Monad (liftM, join)
import Control.Monad.Free (Free(..))
import qualified Control.Monad.Free.Church as CMFC
import Control.Monad.Trans.Except (ExceptT(..), runExceptT)
import           Control.Monad.Trans.Free (FreeF, FreeT(..))
import qualified Control.Monad.Trans.Free as CMTF
import Data.Functor.Identity
import Control.Arrow
import Data.Functor.Compose (Compose(..))
import Data.List.NonEmpty(NonEmpty((:|)), nonEmpty, toList)
import Data.Tree (Tree (..))
#ifdef __GLASGOW_HASKELL__
#if HAS_GENERIC
import GHC.Generics (Generic (..), M1 (..), V1, U1, K1 (..), (:+:) (..), (:*:) (..))
#endif
#endif
import Numeric.Natural
import Prelude

import           Data.Functor.Base hiding (head, tail)
import qualified Data.Functor.Base as NEF (NonEmptyF(..))

import Data.Fix (Fix (..), unFix, Mu (..), Nu (..))

-- $setup
-- >>> :set -XDeriveFunctor -XScopedTypeVariables -XLambdaCase -XGADTs -XFlexibleContexts
-- >>> import Control.Applicative (Const (..), Applicative (..))
-- >>> import Control.Monad (void)
-- >>> import Control.Monad.Trans.Reader (Reader, ask, local, runReader)
-- >>> import Data.Char (toUpper)
-- >>> import Data.Fix (Fix (..))
-- >>> import Data.Foldable (traverse_)
-- >>> import Data.List (intercalate, partition)
-- >>> import Data.List.NonEmpty (NonEmpty (..))
-- >>> import Data.Maybe (maybeToList)
-- >>> import Data.Tree (Tree (..), drawTree)
--
-- >>> import Data.Functor.Base
--
-- >>> let showTree = putStrLn . go where go (Node x xs) = if null xs then x else "(" ++ unwords (x : map go xs) ++ ")"
--
-- >>> let myTree = Node 0 [Node 1 [], Node 2 [], Node 3 [Node 31 [Node 311 [Node 3111 [], Node 3112 []]]]]

-- $foldingFunctions
-- Folding functions allow you to reduce a recursive structure down to a value. The value can be a simple type such as 'Int' or 'String', or it can also be a recursive structure. Each of the functions below will be accompanied by an example which folds the following @Tree Int@ down to some 'String'.
--
-- >>> putStr $ drawTree $ fmap show myTree
-- 0
-- |
-- +- 1
-- |
-- +- 2
-- |
-- `- 3
--    |
--    `- 31
--       |
--       `- 311
--          |
--          +- 3111
--          |
--          `- 3112

-- | Obtain the base functor for a recursive datatype.
--
-- The core idea of this library is that instead of writing recursive functions
-- on a recursive datatype, we prefer to write non-recursive functions on a
-- related, non-recursive datatype we call the "base functor".
--
-- For example, @[a]@ is a recursive type, and its corresponding base functor is
-- @'ListF' a@:
--
-- @
-- data 'ListF' a b = 'Nil' | 'Cons' a b
-- type instance 'Base' [a] = 'ListF' a
-- @
--
-- The relationship between those two types is that if we replace @b@ with
-- @'ListF' a@, we obtain a type which is isomorphic to @[a]@.
--
type family Base t :: * -> *

-- | A recursive datatype which can be unrolled one recursion layer at a time.
--
-- For example, a value of type @[a]@ can be unrolled into a @'ListF' a [a]@.
-- Ifthat unrolled value is a 'Cons', it contains another @[a]@ which can be
-- unrolled as well, and so on.
--
-- Typically, 'Recursive' types also have a 'Corecursive' instance, in which
-- case 'project' and 'embed' are inverses.
class Functor (Base t) => Recursive t where
  -- | Unroll a single recursion layer.
  --
  -- >>> project [1,2,3]
  -- Cons 1 [2,3]
  project :: t -> Base t t
#ifdef HAS_GENERIC
  default project :: (Generic t, Generic (Base t t), GCoerce (Rep t) (Rep (Base t t))) => t -> Base t t
  project = Rep (Base t t) Any -> Base t t
forall a x. Generic a => Rep a x -> a
to (Rep (Base t t) Any -> Base t t)
-> (t -> Rep (Base t t) Any) -> t -> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Rep t Any -> Rep (Base t t) Any
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce (Rep t Any -> Rep (Base t t) Any)
-> (t -> Rep t Any) -> t -> Rep (Base t t) Any
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> Rep t Any
forall a x. Generic a => a -> Rep a x
from
#endif

  -- | An alias for 'fold'.
  --
  -- 'fold' is by far the most common recursion-scheme, because working one layer at a time is the most common strategy for writing a recursive function. But there are also other, rarer strategies. Researchers have given names to the most common strategies, and their name for 'fold' is "catamorphism". They also give its @Base t a -> a@ argument a special name, "(@Base t@)-algebra". More generally, a function of the form @f a -> a@ is called an "f-algebra".
  --
  -- The names might seem intimidating at first, but using the standard nomenclature has benefits. If you program with others, it can be useful to have a shared vocabulary to refer to those recursion patterns. For example, you can discuss which type of recursion is the most appropriate for the problem at hand. Names can also help to structure your thoughts while writing recursive functions.
  --
  -- The rest of this module lists a few of the other recursion-schemes which are common enough to have a name. In this section, we restrict our attention to those which fold a recursive structure down to a value. In the examples all functions will be of type @Tree Int -> String@.
  cata :: (Base t a -> a) -> t -> a
  cata Base t a -> a
f = t -> a
c where c :: t -> a
c = Base t a -> a
f (Base t a -> a) -> (t -> Base t a) -> t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (t -> a) -> Base t t -> Base t a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap t -> a
c (Base t t -> Base t a) -> (t -> Base t t) -> t -> Base t a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> Base t t
forall t. Recursive t => t -> Base t t
project

  -- | A variant of 'cata' in which recursive positions also include the
  -- original sub-tree, in addition to the result of folding that sub-tree.
  --
  -- For our running example, let's add a number to each node indicating how
  -- many children are below it. To do so, we will need to count those nodes
  -- from the original sub-tree.
  --
  -- >>> :{
  -- let pprint4 :: Tree Int -> String
  --     pprint4 = flip runReader 0 . para go
  --       where
  --         go :: TreeF Int (Tree Int, Reader Int String)
  --            -> Reader Int String
  --         go (NodeF i trss) = do
  --           -- trss :: [(Tree Int, Reader Int String)]
  --           -- ts   :: [Tree Int]
  --           -- rss  :: [Reader Int String]
  --           -- ss   :: [String]
  --           let (ts, rss) = unzip trss
  --           let count = sum $ fmap length ts
  --           ss <- local (+ 2) $ sequence rss
  --           indent <- ask
  --           let s = replicate indent ' '
  --                ++ "* " ++ show i
  --                ++ " (" ++ show count ++ ")"
  --           pure $ intercalate "\n" (s : ss)
  -- :}
  --
  -- >>> putStrLn $ pprint4 myTree
  -- * 0 (7)
  --   * 1 (0)
  --   * 2 (0)
  --   * 3 (4)
  --     * 31 (3)
  --       * 311 (2)
  --         * 3111 (0)
  --         * 3112 (0)
  --
  -- One common use for 'para' is to construct a new tree which reuses most of
  -- the sub-trees from the original. In the following example, we insert a new
  -- node under the leftmost leaf. This requires allocating new nodes along a
  -- path from the root to that leaf, while keeping every other sub-tree
  -- untouched.
  --
  -- >>> :{
  -- let insertLeftmost :: Int -> Tree Int -> Tree Int
  --     insertLeftmost new = para go
  --       where
  --         go :: TreeF Int (Tree Int, Tree Int)
  --            -> Tree Int
  --         go (NodeF i []) = Node i [Node new []]
  --         go (NodeF i ((_orig, recur) : tts)) =
  --           -- tts :: [(Tree Int, Tree Int)]
  --           let (origs, _recurs) = unzip tts
  --           in Node i (recur : origs)
  -- :}
  --
  -- >>> putStrLn $ pprint4 $ insertLeftmost 999 myTree
  -- * 0 (8)
  --   * 1 (1)
  --     * 999 (0)
  --   * 2 (0)
  --   * 3 (4)
  --     * 31 (3)
  --       * 311 (2)
  --         * 3111 (0)
  --         * 3112 (0)
  para :: (Base t (t, a) -> a) -> t -> a
  para Base t (t, a) -> a
t = t -> a
p where p :: t -> a
p t
x = Base t (t, a) -> a
t (Base t (t, a) -> a)
-> (Base t t -> Base t (t, a)) -> Base t t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (t -> (t, a)) -> Base t t -> Base t (t, a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((,) (t -> a -> (t, a)) -> (t -> a) -> t -> (t, a)
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> t -> a
p) (Base t t -> a) -> Base t t -> a
forall a b. (a -> b) -> a -> b
$ t -> Base t t
forall t. Recursive t => t -> Base t t
project t
x

  gpara :: (Corecursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (EnvT t w a) -> a) -> t -> a
  gpara forall b. Base t (w b) -> w (Base t b)
t = (Base t t -> t)
-> (forall b. Base t (w b) -> w (Base t b))
-> (Base t (EnvT t w a) -> a)
-> t
-> a
forall t (w :: * -> *) b a.
(Recursive t, Comonad w) =>
(Base t b -> b)
-> (forall c. Base t (w c) -> w (Base t c))
-> (Base t (EnvT b w a) -> a)
-> t
-> a
gzygo Base t t -> t
forall t. Corecursive t => Base t t -> t
embed forall b. Base t (w b) -> w (Base t b)
t

  -- | Fokkinga's prepromorphism
  prepro
    :: Corecursive t
    => (forall b. Base t b -> Base t b)
    -> (Base t a -> a)
    -> t
    -> a
  prepro forall b. Base t b -> Base t b
e Base t a -> a
f = t -> a
c where c :: t -> a
c = Base t a -> a
f (Base t a -> a) -> (t -> Base t a) -> t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (t -> a) -> Base t t -> Base t a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (t -> a
c (t -> a) -> (t -> t) -> t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall b. Base t b -> Base t b) -> t -> t
forall s t.
(Recursive s, Corecursive t) =>
(forall a. Base s a -> Base t a) -> s -> t
hoist forall b. Base t b -> Base t b
e) (Base t t -> Base t a) -> (t -> Base t t) -> t -> Base t a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> Base t t
forall t. Recursive t => t -> Base t t
project

  --- | A generalized prepromorphism
  gprepro
    :: (Corecursive t, Comonad w)
    => (forall b. Base t (w b) -> w (Base t b))
    -> (forall c. Base t c -> Base t c)
    -> (Base t (w a) -> a)
    -> t
    -> a
  gprepro forall b. Base t (w b) -> w (Base t b)
k forall b. Base t b -> Base t b
e Base t (w a) -> a
f = w a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract (w a -> a) -> (t -> w a) -> t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> w a
c where c :: t -> w a
c = (Base t (w a) -> a) -> w (Base t (w a)) -> w a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Base t (w a) -> a
f (w (Base t (w a)) -> w a) -> (t -> w (Base t (w a))) -> t -> w a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Base t (w (w a)) -> w (Base t (w a))
forall b. Base t (w b) -> w (Base t b)
k (Base t (w (w a)) -> w (Base t (w a)))
-> (t -> Base t (w (w a))) -> t -> w (Base t (w a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (t -> w (w a)) -> Base t t -> Base t (w (w a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (w a -> w (w a)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate (w a -> w (w a)) -> (t -> w a) -> t -> w (w a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> w a
c (t -> w a) -> (t -> t) -> t -> w a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall b. Base t b -> Base t b) -> t -> t
forall s t.
(Recursive s, Corecursive t) =>
(forall a. Base s a -> Base t a) -> s -> t
hoist forall b. Base t b -> Base t b
e) (Base t t -> Base t (w (w a)))
-> (t -> Base t t) -> t -> Base t (w (w a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> Base t t
forall t. Recursive t => t -> Base t t
project

distPara :: Corecursive t => Base t (t, a) -> (t, Base t a)
distPara :: forall t a. Corecursive t => Base t (t, a) -> (t, Base t a)
distPara = (Base t t -> t) -> Base t (t, a) -> (t, Base t a)
forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> f (b, a) -> (b, f a)
distZygo Base t t -> t
forall t. Corecursive t => Base t t -> t
embed

distParaT :: (Corecursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> Base t (EnvT t w a) -> EnvT t w (Base t a)
distParaT :: forall t (w :: * -> *) a.
(Corecursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> Base t (EnvT t w a) -> EnvT t w (Base t a)
distParaT forall b. Base t (w b) -> w (Base t b)
t = (Base t t -> t)
-> (forall b. Base t (w b) -> w (Base t b))
-> Base t (EnvT t w a)
-> EnvT t w (Base t a)
forall (f :: * -> *) (w :: * -> *) b a.
(Functor f, Comonad w) =>
(f b -> b)
-> (forall c. f (w c) -> w (f c))
-> f (EnvT b w a)
-> EnvT b w (f a)
distZygoT Base t t -> t
forall t. Corecursive t => Base t t -> t
embed forall b. Base t (w b) -> w (Base t b)
t

-- | A recursive datatype which can be rolled up one recursion layer at a time.
--
-- For example, a value of type @'ListF' a [a]@ can be rolled up into a @[a]@.
-- This @[a]@ can then be used in a 'Cons' to construct another @'ListF' a [a]@,
-- which can be rolled up as well, and so on.
--
-- Typically, 'Corecursive' types also have a 'Recursive' instance, in which
-- case 'embed' and 'project' are inverses.
class Functor (Base t) => Corecursive t where

  -- | Roll up a single recursion layer.
  --
  -- >>> embed (Cons 1 [2,3])
  -- [1,2,3]
  embed :: Base t t -> t
#ifdef HAS_GENERIC
  default embed :: (Generic t, Generic (Base t t), GCoerce (Rep (Base t t)) (Rep t)) => Base t t -> t
  embed = Rep t Any -> t
forall a x. Generic a => Rep a x -> a
to (Rep t Any -> t) -> (Base t t -> Rep t Any) -> Base t t -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Rep (Base t t) Any -> Rep t Any
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce (Rep (Base t t) Any -> Rep t Any)
-> (Base t t -> Rep (Base t t) Any) -> Base t t -> Rep t Any
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Base t t -> Rep (Base t t) Any
forall a x. Generic a => a -> Rep a x
from
#endif

  -- | An alias for 'unfold'.
  ana
    :: (a -> Base t a) -- ^ a (Base t)-coalgebra
    -> a               -- ^ seed
    -> t               -- ^ resulting fixed point
  ana a -> Base t a
g = a -> t
a where a :: a -> t
a = Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t) -> (a -> Base t t) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> t) -> Base t a -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> t
a (Base t a -> Base t t) -> (a -> Base t a) -> a -> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Base t a
g

  apo :: (a -> Base t (Either t a)) -> a -> t
  apo a -> Base t (Either t a)
g = a -> t
a where a :: a -> t
a = Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t) -> (a -> Base t t) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ((Either t a -> t) -> Base t (Either t a) -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((t -> t) -> (a -> t) -> Either t a -> t
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either t -> t
forall a. a -> a
id a -> t
a)) (Base t (Either t a) -> Base t t)
-> (a -> Base t (Either t a)) -> a -> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Base t (Either t a)
g

  -- | Fokkinga's postpromorphism
  postpro
    :: Recursive t
    => (forall b. Base t b -> Base t b) -- natural transformation
    -> (a -> Base t a)                  -- a (Base t)-coalgebra
    -> a                                -- seed
    -> t
  postpro forall b. Base t b -> Base t b
e a -> Base t a
g = a -> t
a where a :: a -> t
a = Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t) -> (a -> Base t t) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> t) -> Base t a -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((forall b. Base t b -> Base t b) -> t -> t
forall s t.
(Recursive s, Corecursive t) =>
(forall a. Base s a -> Base t a) -> s -> t
hoist forall b. Base t b -> Base t b
e (t -> t) -> (a -> t) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> t
a) (Base t a -> Base t t) -> (a -> Base t a) -> a -> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Base t a
g

  -- | A generalized postpromorphism
  gpostpro
    :: (Recursive t, Monad m)
    => (forall b. m (Base t b) -> Base t (m b)) -- distributive law
    -> (forall c. Base t c -> Base t c)         -- natural transformation
    -> (a -> Base t (m a))                      -- a (Base t)-m-coalgebra
    -> a                                        -- seed
    -> t
  gpostpro forall b. m (Base t b) -> Base t (m b)
k forall b. Base t b -> Base t b
e a -> Base t (m a)
g = m a -> t
a (m a -> t) -> (a -> m a) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> m a
forall (m :: * -> *) a. Monad m => a -> m a
return where a :: m a -> t
a = Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t) -> (m a -> Base t t) -> m a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (m (m a) -> t) -> Base t (m (m a)) -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((forall b. Base t b -> Base t b) -> t -> t
forall s t.
(Recursive s, Corecursive t) =>
(forall a. Base s a -> Base t a) -> s -> t
hoist forall b. Base t b -> Base t b
e (t -> t) -> (m (m a) -> t) -> m (m a) -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m a -> t
a (m a -> t) -> (m (m a) -> m a) -> m (m a) -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (m a) -> m a
forall (m :: * -> *) a. Monad m => m (m a) -> m a
join) (Base t (m (m a)) -> Base t t)
-> (m a -> Base t (m (m a))) -> m a -> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (Base t (m a)) -> Base t (m (m a))
forall b. m (Base t b) -> Base t (m b)
k (m (Base t (m a)) -> Base t (m (m a)))
-> (m a -> m (Base t (m a))) -> m a -> Base t (m (m a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Base t (m a)) -> m a -> m (Base t (m a))
forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM a -> Base t (m a)
g

-- | An alias for 'refold'.
hylo :: Functor f => (f b -> b) -> (a -> f a) -> a -> b
hylo :: forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> (a -> f a) -> a -> b
hylo f b -> b
f a -> f a
g = a -> b
h where h :: a -> b
h = f b -> b
f (f b -> b) -> (a -> f b) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
h (f a -> f b) -> (a -> f a) -> a -> f b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> f a
g

-- | Folds a recursive type down to a value, one layer at a time.
--
-- >>> :{
-- let mySum :: [Int] -> Int
--     mySum = fold $ \case
--       Nil -> 0
--       Cons x sumXs -> x + sumXs
-- :}
--
-- >>> mySum [10,11,12]
-- 33
--
-- In our running example, one layer consists of an 'Int' and a list of recursive positions. In @Tree Int@, those recursive positions contain sub-trees of type @Tree Int@. Since we are working one layer at a time, the @Base t a -> a@ function is not given a @Tree Int@, but a @TreeF Int String@. That is, each recursive position contains the 'String' resulting from recursively folding the corresponding sub-tree.
--
-- >>> :{
-- let pprint1 :: Tree Int -> String
--     pprint1 = fold $ \case
--       NodeF i [] -> show i
--       NodeF i ss -> show i ++ ": [" ++ intercalate ", " ss ++ "]"
-- :}
--
-- >>> putStrLn $ pprint1 myTree
-- 0: [1, 2, 3: [31: [311: [3111, 3112]]]]
--
-- More generally, the 't' argument is the recursive value, the 'a' is the final result, and the @Base t a -> a@ function explains how to reduce a single layer full of recursive results down to a result.
fold :: Recursive t => (Base t a -> a) -> t -> a
fold :: forall t a. Recursive t => (Base t a -> a) -> t -> a
fold = (Base t a -> a) -> t -> a
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata

-- | A generalization of 'unfoldr'. The starting seed is expanded into a base
-- functor whose recursive positions contain more seeds, which are themselves
-- expanded, and so on.
--
-- >>> :{
-- >>> let ourEnumFromTo :: Int -> Int -> [Int]
-- >>>     ourEnumFromTo lo hi = ana go lo where
-- >>>         go i = if i > hi then Nil else Cons i (i + 1)
-- >>> :}
--
-- >>> ourEnumFromTo 1 4
-- [1,2,3,4]
unfold :: Corecursive t => (a -> Base t a) -> a -> t
unfold :: forall t a. Corecursive t => (a -> Base t a) -> a -> t
unfold = (a -> Base t a) -> a -> t
forall t a. Corecursive t => (a -> Base t a) -> a -> t
ana

-- | An optimized version of @fold f . unfold g@.
--
-- Useful when your recursion structure is shaped like a particular recursive
-- datatype, but you're neither consuming nor producing that recursive datatype.
-- For example, the recursion structure of quick sort is a binary tree, but its
-- input and output is a list, not a binary tree.
--
-- >>> data BinTreeF a b = Tip | Branch b a b deriving (Functor)
--
-- >>> :{
-- >>> let quicksort :: Ord a => [a] -> [a]
-- >>>     quicksort = refold merge split where
-- >>>         split []     = Tip
-- >>>         split (x:xs) = let (l, r) = partition (<x) xs in Branch l x r
-- >>>
-- >>>         merge Tip            = []
-- >>>         merge (Branch l x r) = l ++ [x] ++ r
-- >>> :}
--
-- >>> quicksort [1,5,2,8,4,9,8]
-- [1,2,4,5,8,8,9]
refold :: Functor f => (f b -> b) -> (a -> f a) -> a -> b
refold :: forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> (a -> f a) -> a -> b
refold = (f b -> b) -> (a -> f a) -> a -> b
forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> (a -> f a) -> a -> b
hylo

type instance Base [a] = ListF a
instance Recursive [a] where
  project :: [a] -> Base [a] [a]
project (a
x:[a]
xs) = a -> [a] -> ListF a [a]
forall a b. a -> b -> ListF a b
Cons a
x [a]
xs
  project [] = Base [a] [a]
forall a b. ListF a b
Nil

  para :: forall a. (Base [a] ([a], a) -> a) -> [a] -> a
para Base [a] ([a], a) -> a
f (a
x:[a]
xs) = Base [a] ([a], a) -> a
f (a -> ([a], a) -> ListF a ([a], a)
forall a b. a -> b -> ListF a b
Cons a
x ([a]
xs, (Base [a] ([a], a) -> a) -> [a] -> a
forall t a. Recursive t => (Base t (t, a) -> a) -> t -> a
para Base [a] ([a], a) -> a
f [a]
xs))
  para Base [a] ([a], a) -> a
f [] = Base [a] ([a], a) -> a
f Base [a] ([a], a)
forall a b. ListF a b
Nil

instance Corecursive [a] where
  embed :: Base [a] [a] -> [a]
embed (Cons a
x [a]
xs) = a
xa -> [a] -> [a]
forall a. a -> [a] -> [a]
:[a]
xs
  embed ListF a [a]
Base [a] [a]
Nil = []

  apo :: forall a. (a -> Base [a] (Either [a] a)) -> a -> [a]
apo a -> Base [a] (Either [a] a)
f a
a = case a -> Base [a] (Either [a] a)
f a
a of
    Cons a
x (Left [a]
xs) -> a
x a -> [a] -> [a]
forall a. a -> [a] -> [a]
: [a]
xs
    Cons a
x (Right a
b) -> a
x a -> [a] -> [a]
forall a. a -> [a] -> [a]
: (a -> Base [a] (Either [a] a)) -> a -> [a]
forall t a. Corecursive t => (a -> Base t (Either t a)) -> a -> t
apo a -> Base [a] (Either [a] a)
f a
b
    ListF a (Either [a] a)
Base [a] (Either [a] a)
Nil -> []

type instance Base (NonEmpty a) = NonEmptyF a
instance Recursive (NonEmpty a) where
  project :: NonEmpty a -> Base (NonEmpty a) (NonEmpty a)
project (a
x:|[a]
xs) = a -> Maybe (NonEmpty a) -> NonEmptyF a (NonEmpty a)
forall a b. a -> Maybe b -> NonEmptyF a b
NonEmptyF a
x (Maybe (NonEmpty a) -> NonEmptyF a (NonEmpty a))
-> Maybe (NonEmpty a) -> NonEmptyF a (NonEmpty a)
forall a b. (a -> b) -> a -> b
$ [a] -> Maybe (NonEmpty a)
forall a. [a] -> Maybe (NonEmpty a)
nonEmpty [a]
xs
instance Corecursive (NonEmpty a) where
  embed :: Base (NonEmpty a) (NonEmpty a) -> NonEmpty a
embed = a -> [a] -> NonEmpty a
forall a. a -> [a] -> NonEmpty a
(:|) (a -> [a] -> NonEmpty a)
-> (NonEmptyF a (NonEmpty a) -> a)
-> NonEmptyF a (NonEmpty a)
-> [a]
-> NonEmpty a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> NonEmptyF a (NonEmpty a) -> a
forall a b. NonEmptyF a b -> a
NEF.head (NonEmptyF a (NonEmpty a) -> [a] -> NonEmpty a)
-> (NonEmptyF a (NonEmpty a) -> [a])
-> NonEmptyF a (NonEmpty a)
-> NonEmpty a
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> ([a] -> (NonEmpty a -> [a]) -> Maybe (NonEmpty a) -> [a]
forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] NonEmpty a -> [a]
forall a. NonEmpty a -> [a]
toList (Maybe (NonEmpty a) -> [a])
-> (NonEmptyF a (NonEmpty a) -> Maybe (NonEmpty a))
-> NonEmptyF a (NonEmpty a)
-> [a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> NonEmptyF a (NonEmpty a) -> Maybe (NonEmpty a)
forall a b. NonEmptyF a b -> Maybe b
NEF.tail)

type instance Base (Tree a) = TreeF a
instance Recursive (Tree a) where
  project :: Tree a -> Base (Tree a) (Tree a)
project (Node a
x [Tree a]
xs) = a -> [Tree a] -> TreeF a (Tree a)
forall a b. a -> ForestF a b -> TreeF a b
NodeF a
x [Tree a]
xs
instance Corecursive (Tree a) where
  embed :: Base (Tree a) (Tree a) -> Tree a
embed (NodeF a
x ForestF a (Tree a)
xs) = a -> ForestF a (Tree a) -> Tree a
forall a. a -> [Tree a] -> Tree a
Node a
x ForestF a (Tree a)
xs

type instance Base Natural = Maybe
instance Recursive Natural where
  project :: Natural -> Base Natural Natural
project Natural
0 = Base Natural Natural
forall a. Maybe a
Nothing
  project Natural
n = Natural -> Maybe Natural
forall a. a -> Maybe a
Just (Natural
n Natural -> Natural -> Natural
forall a. Num a => a -> a -> a
- Natural
1)
instance Corecursive Natural where
  embed :: Base Natural Natural -> Natural
embed = Natural -> (Natural -> Natural) -> Maybe Natural -> Natural
forall b a. b -> (a -> b) -> Maybe a -> b
maybe Natural
0 (Natural -> Natural -> Natural
forall a. Num a => a -> a -> a
+Natural
1)

-- | Cofree comonads are Recursive/Corecursive
type instance Base (Cofree f a) = CofreeF f a
instance Functor f => Recursive (Cofree f a) where
  project :: Cofree f a -> Base (Cofree f a) (Cofree f a)
project (a
x :< f (Cofree f a)
xs) = a
x a -> f (Cofree f a) -> CofreeF f a (Cofree f a)
forall (f :: * -> *) a b. a -> f b -> CofreeF f a b
CCTC.:< f (Cofree f a)
xs
instance Functor f => Corecursive (Cofree f a) where
  embed :: Base (Cofree f a) (Cofree f a) -> Cofree f a
embed (a
x CCTC.:< f (Cofree f a)
xs) = a
x a -> f (Cofree f a) -> Cofree f a
forall (f :: * -> *) a. a -> f (Cofree f a) -> Cofree f a
:< f (Cofree f a)
xs

-- | Cofree tranformations of comonads are Recursive/Corecusive
type instance Base (CofreeT f w a) = Compose w (CofreeF f a)
instance (Functor w, Functor f) => Recursive (CofreeT f w a) where
  project :: CofreeT f w a -> Base (CofreeT f w a) (CofreeT f w a)
project = w (CofreeF f a (CofreeT f w a))
-> Compose w (CofreeF f a) (CofreeT f w a)
forall {k} {k1} (f :: k -> *) (g :: k1 -> k) (a :: k1).
f (g a) -> Compose f g a
Compose (w (CofreeF f a (CofreeT f w a))
 -> Compose w (CofreeF f a) (CofreeT f w a))
-> (CofreeT f w a -> w (CofreeF f a (CofreeT f w a)))
-> CofreeT f w a
-> Compose w (CofreeF f a) (CofreeT f w a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. CofreeT f w a -> w (CofreeF f a (CofreeT f w a))
forall (f :: * -> *) (w :: * -> *) a.
CofreeT f w a -> w (CofreeF f a (CofreeT f w a))
runCofreeT
instance (Functor w, Functor f) => Corecursive (CofreeT f w a) where
  embed :: Base (CofreeT f w a) (CofreeT f w a) -> CofreeT f w a
embed = w (CofreeF f a (CofreeT f w a)) -> CofreeT f w a
forall (f :: * -> *) (w :: * -> *) a.
w (CofreeF f a (CofreeT f w a)) -> CofreeT f w a
CofreeT (w (CofreeF f a (CofreeT f w a)) -> CofreeT f w a)
-> (Compose w (CofreeF f a) (CofreeT f w a)
    -> w (CofreeF f a (CofreeT f w a)))
-> Compose w (CofreeF f a) (CofreeT f w a)
-> CofreeT f w a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Compose w (CofreeF f a) (CofreeT f w a)
-> w (CofreeF f a (CofreeT f w a))
forall {k1} {k2} (f :: k1 -> *) (g :: k2 -> k1) (a :: k2).
Compose f g a -> f (g a)
getCompose

-- | Free monads are Recursive/Corecursive
type instance Base (Free f a) = FreeF f a

instance Functor f => Recursive (Free f a) where
  project :: Free f a -> Base (Free f a) (Free f a)
project (Pure a
a) = a -> FreeF f a (Free f a)
forall (f :: * -> *) a b. a -> FreeF f a b
CMTF.Pure a
a
  project (Free f (Free f a)
f) = f (Free f a) -> FreeF f a (Free f a)
forall (f :: * -> *) a b. f b -> FreeF f a b
CMTF.Free f (Free f a)
f

improveF :: Functor f => CMFC.F f a -> Free f a
improveF :: forall (f :: * -> *) a. Functor f => F f a -> Free f a
improveF F f a
x = (forall (m :: * -> *). MonadFree f m => m a) -> Free f a
forall (f :: * -> *) a.
Functor f =>
(forall (m :: * -> *). MonadFree f m => m a) -> Free f a
CMFC.improve (F f a -> m a
forall (f :: * -> *) (m :: * -> *) a. MonadFree f m => F f a -> m a
CMFC.fromF F f a
x)
-- | It may be better to work with the instance for `CMFC.F` directly.
instance Functor f => Corecursive (Free f a) where
  embed :: Base (Free f a) (Free f a) -> Free f a
embed (CMTF.Pure a
a) = a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure a
a
  embed (CMTF.Free f (Free f a)
f) = f (Free f a) -> Free f a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free f (Free f a)
f
  ana :: forall a. (a -> Base (Free f a) a) -> a -> Free f a
ana               a -> Base (Free f a) a
coalg = F f a -> Free f a
forall (f :: * -> *) a. Functor f => F f a -> Free f a
improveF (F f a -> Free f a) -> (a -> F f a) -> a -> Free f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Base (F f a) a) -> a -> F f a
forall t a. Corecursive t => (a -> Base t a) -> a -> t
ana               a -> Base (F f a) a
a -> Base (Free f a) a
coalg
  postpro :: forall a.
Recursive (Free f a) =>
(forall b. Base (Free f a) b -> Base (Free f a) b)
-> (a -> Base (Free f a) a) -> a -> Free f a
postpro       forall b. Base (Free f a) b -> Base (Free f a) b
nat a -> Base (Free f a) a
coalg = F f a -> Free f a
forall (f :: * -> *) a. Functor f => F f a -> Free f a
improveF (F f a -> Free f a) -> (a -> F f a) -> a -> Free f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall b. Base (F f a) b -> Base (F f a) b)
-> (a -> Base (F f a) a) -> a -> F f a
forall t a.
(Corecursive t, Recursive t) =>
(forall b. Base t b -> Base t b) -> (a -> Base t a) -> a -> t
postpro       forall b. Base (F f a) b -> Base (F f a) b
forall b. Base (Free f a) b -> Base (Free f a) b
nat a -> Base (F f a) a
a -> Base (Free f a) a
coalg
  gpostpro :: forall (m :: * -> *) a.
(Recursive (Free f a), Monad m) =>
(forall b. m (Base (Free f a) b) -> Base (Free f a) (m b))
-> (forall b. Base (Free f a) b -> Base (Free f a) b)
-> (a -> Base (Free f a) (m a))
-> a
-> Free f a
gpostpro forall b. m (Base (Free f a) b) -> Base (Free f a) (m b)
dist forall b. Base (Free f a) b -> Base (Free f a) b
nat a -> Base (Free f a) (m a)
coalg = F f a -> Free f a
forall (f :: * -> *) a. Functor f => F f a -> Free f a
improveF (F f a -> Free f a) -> (a -> F f a) -> a -> Free f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall b. m (Base (F f a) b) -> Base (F f a) (m b))
-> (forall b. Base (F f a) b -> Base (F f a) b)
-> (a -> Base (F f a) (m a))
-> a
-> F f a
forall t (m :: * -> *) a.
(Corecursive t, Recursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (forall b. Base t b -> Base t b)
-> (a -> Base t (m a))
-> a
-> t
gpostpro forall b. m (Base (F f a) b) -> Base (F f a) (m b)
forall b. m (Base (Free f a) b) -> Base (Free f a) (m b)
dist forall b. Base (F f a) b -> Base (F f a) b
forall b. Base (Free f a) b -> Base (Free f a) b
nat a -> Base (F f a) (m a)
a -> Base (Free f a) (m a)
coalg

-- | Free transformations of monads are Recursive/Corecursive
type instance Base (FreeT f m a) = Compose m (FreeF f a)
instance (Functor m, Functor f) => Recursive (FreeT f m a) where
  project :: FreeT f m a -> Base (FreeT f m a) (FreeT f m a)
project = m (FreeF f a (FreeT f m a)) -> Compose m (FreeF f a) (FreeT f m a)
forall {k} {k1} (f :: k -> *) (g :: k1 -> k) (a :: k1).
f (g a) -> Compose f g a
Compose (m (FreeF f a (FreeT f m a))
 -> Compose m (FreeF f a) (FreeT f m a))
-> (FreeT f m a -> m (FreeF f a (FreeT f m a)))
-> FreeT f m a
-> Compose m (FreeF f a) (FreeT f m a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FreeT f m a -> m (FreeF f a (FreeT f m a))
forall (f :: * -> *) (m :: * -> *) a.
FreeT f m a -> m (FreeF f a (FreeT f m a))
runFreeT
instance (Functor m, Functor f) => Corecursive (FreeT f m a) where
  embed :: Base (FreeT f m a) (FreeT f m a) -> FreeT f m a
embed = 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 (m (FreeF f a (FreeT f m a)) -> FreeT f m a)
-> (Compose m (FreeF f a) (FreeT f m a)
    -> m (FreeF f a (FreeT f m a)))
-> Compose m (FreeF f a) (FreeT f m a)
-> FreeT f m a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Compose m (FreeF f a) (FreeT f m a) -> m (FreeF f a (FreeT f m a))
forall {k1} {k2} (f :: k1 -> *) (g :: k2 -> k1) (a :: k2).
Compose f g a -> f (g a)
getCompose

-- If you are looking for instances for the free MonadPlus, please use the
-- instance for FreeT f [].

-- If you are looking for instances for the free alternative and free
-- applicative, I'm sorry to disapoint you but you won't find them in this
-- package.  They can be considered recurive, but using non-uniform recursion;
-- this package only implements uniformly recursive folds / unfolds.

-- | Example boring stub for non-recursive data types
type instance Base (Maybe a) = Const (Maybe a)
instance Recursive (Maybe a) where project :: Maybe a -> Base (Maybe a) (Maybe a)
project = Maybe a -> Base (Maybe a) (Maybe a)
forall {k} a (b :: k). a -> Const a b
Const
instance Corecursive (Maybe a) where embed :: Base (Maybe a) (Maybe a) -> Maybe a
embed = Base (Maybe a) (Maybe a) -> Maybe a
forall {k} a (b :: k). Const a b -> a
getConst

-- | Example boring stub for non-recursive data types
type instance Base (Either a b) = Const (Either a b)
instance Recursive (Either a b) where project :: Either a b -> Base (Either a b) (Either a b)
project = Either a b -> Base (Either a b) (Either a b)
forall {k} a (b :: k). a -> Const a b
Const
instance Corecursive (Either a b) where embed :: Base (Either a b) (Either a b) -> Either a b
embed = Base (Either a b) (Either a b) -> Either a b
forall {k} a (b :: k). Const a b -> a
getConst

-- | A generalized catamorphism
gfold, gcata
  :: (Recursive t, Comonad w)
  => (forall b. Base t (w b) -> w (Base t b)) -- ^ a distributive law
  -> (Base t (w a) -> a)                      -- ^ a (Base t)-w-algebra
  -> t                                        -- ^ fixed point
  -> a
gcata :: forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gcata forall b. Base t (w b) -> w (Base t b)
k Base t (w a) -> a
g = Base t (w a) -> a
g (Base t (w a) -> a) -> (t -> Base t (w a)) -> t -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. w (Base t (w a)) -> Base t (w a)
forall (w :: * -> *) a. Comonad w => w a -> a
extract (w (Base t (w a)) -> Base t (w a))
-> (t -> w (Base t (w a))) -> t -> Base t (w a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> w (Base t (w a))
c where
  c :: t -> w (Base t (w a))
c = Base t (w (w a)) -> w (Base t (w a))
forall b. Base t (w b) -> w (Base t b)
k (Base t (w (w a)) -> w (Base t (w a)))
-> (t -> Base t (w (w a))) -> t -> w (Base t (w a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (t -> w (w a)) -> Base t t -> Base t (w (w a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (w a -> w (w a)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate (w a -> w (w a)) -> (t -> w a) -> t -> w (w a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Base t (w a) -> a) -> w (Base t (w a)) -> w a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Base t (w a) -> a
g (w (Base t (w a)) -> w a) -> (t -> w (Base t (w a))) -> t -> w a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> w (Base t (w a))
c) (Base t t -> Base t (w (w a)))
-> (t -> Base t t) -> t -> Base t (w (w a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. t -> Base t t
forall t. Recursive t => t -> Base t t
project
gfold :: forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gfold forall b. Base t (w b) -> w (Base t b)
k Base t (w a) -> a
g t
t = (forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gcata forall b. Base t (w b) -> w (Base t b)
k Base t (w a) -> a
g t
t

distCata :: Functor f => f (Identity a) -> Identity (f a)
distCata :: forall (f :: * -> *) a.
Functor f =>
f (Identity a) -> Identity (f a)
distCata = f a -> Identity (f a)
forall a. a -> Identity a
Identity (f a -> Identity (f a))
-> (f (Identity a) -> f a) -> f (Identity a) -> Identity (f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Identity a -> a) -> f (Identity a) -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Identity a -> a
forall a. Identity a -> a
runIdentity

-- | A generalized anamorphism
gunfold, gana
  :: (Corecursive t, Monad m)
  => (forall b. m (Base t b) -> Base t (m b)) -- ^ a distributive law
  -> (a -> Base t (m a))                      -- ^ a (Base t)-m-coalgebra
  -> a                                        -- ^ seed
  -> t
gana :: forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gana forall b. m (Base t b) -> Base t (m b)
k a -> Base t (m a)
f = m (Base t (m a)) -> t
a (m (Base t (m a)) -> t) -> (a -> m (Base t (m a))) -> a -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Base t (m a) -> m (Base t (m a))
forall (m :: * -> *) a. Monad m => a -> m a
return (Base t (m a) -> m (Base t (m a)))
-> (a -> Base t (m a)) -> a -> m (Base t (m a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Base t (m a)
f where
  a :: m (Base t (m a)) -> t
a = Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t)
-> (m (Base t (m a)) -> Base t t) -> m (Base t (m a)) -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (m (m a) -> t) -> Base t (m (m a)) -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (m (Base t (m a)) -> t
a (m (Base t (m a)) -> t)
-> (m (m a) -> m (Base t (m a))) -> m (m a) -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> Base t (m a)) -> m a -> m (Base t (m a))
forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM a -> Base t (m a)
f (m a -> m (Base t (m a)))
-> (m (m a) -> m a) -> m (m a) -> m (Base t (m a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (m a) -> m a
forall (m :: * -> *) a. Monad m => m (m a) -> m a
join) (Base t (m (m a)) -> Base t t)
-> (m (Base t (m a)) -> Base t (m (m a)))
-> m (Base t (m a))
-> Base t t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (Base t (m a)) -> Base t (m (m a))
forall b. m (Base t b) -> Base t (m b)
k
gunfold :: forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gunfold forall b. m (Base t b) -> Base t (m b)
k a -> Base t (m a)
f a
t = (forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gana forall b. m (Base t b) -> Base t (m b)
k a -> Base t (m a)
f a
t

distAna :: Functor f => Identity (f a) -> f (Identity a)
distAna :: forall (f :: * -> *) a.
Functor f =>
Identity (f a) -> f (Identity a)
distAna = (a -> Identity a) -> f a -> f (Identity a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> Identity a
forall a. a -> Identity a
Identity (f a -> f (Identity a))
-> (Identity (f a) -> f a) -> Identity (f a) -> f (Identity a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Identity (f a) -> f a
forall a. Identity a -> a
runIdentity

-- | A generalized hylomorphism
grefold, ghylo
  :: (Comonad w, Functor f, Monad m)
  => (forall c. f (w c) -> w (f c))
  -> (forall d. m (f d) -> f (m d))
  -> (f (w b) -> b)
  -> (a -> f (m a))
  -> a
  -> b
ghylo :: forall (w :: * -> *) (f :: * -> *) (m :: * -> *) b a.
(Comonad w, Functor f, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
ghylo forall c. f (w c) -> w (f c)
w forall d. m (f d) -> f (m d)
m f (w b) -> b
f a -> f (m a)
g = f (w b) -> b
f (f (w b) -> b) -> (a -> f (w b)) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (m a -> w b) -> f (m a) -> f (w b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((f (w b) -> w b) -> (m a -> f (m a)) -> m a -> w b
forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> (a -> f a) -> a -> b
hylo f (w b) -> w b
alg m a -> f (m a)
coalg) (f (m a) -> f (w b)) -> (a -> f (m a)) -> a -> f (w b)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> f (m a)
g where
  coalg :: m a -> f (m a)
coalg = (m (m a) -> m a) -> f (m (m a)) -> f (m a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap m (m a) -> m a
forall (m :: * -> *) a. Monad m => m (m a) -> m a
join (f (m (m a)) -> f (m a)) -> (m a -> f (m (m a))) -> m a -> f (m a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (f (m a)) -> f (m (m a))
forall d. m (f d) -> f (m d)
m (m (f (m a)) -> f (m (m a)))
-> (m a -> m (f (m a))) -> m a -> f (m (m a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> f (m a)) -> m a -> m (f (m a))
forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM a -> f (m a)
g
  alg :: f (w b) -> w b
alg   = (f (w b) -> b) -> w (f (w b)) -> w b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap f (w b) -> b
f (w (f (w b)) -> w b) -> (f (w b) -> w (f (w b))) -> f (w b) -> w b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. f (w (w b)) -> w (f (w b))
forall c. f (w c) -> w (f c)
w (f (w (w b)) -> w (f (w b)))
-> (f (w b) -> f (w (w b))) -> f (w b) -> w (f (w b))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (w b -> w (w b)) -> f (w b) -> f (w (w b))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap w b -> w (w b)
forall (w :: * -> *) a. Comonad w => w a -> w (w a)
duplicate
grefold :: forall (w :: * -> *) (f :: * -> *) (m :: * -> *) b a.
(Comonad w, Functor f, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
grefold forall c. f (w c) -> w (f c)
w forall d. m (f d) -> f (m d)
m f (w b) -> b
f a -> f (m a)
g a
a = (forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
forall (w :: * -> *) (f :: * -> *) (m :: * -> *) b a.
(Comonad w, Functor f, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
ghylo forall c. f (w c) -> w (f c)
w forall d. m (f d) -> f (m d)
m f (w b) -> b
f a -> f (m a)
g a
a

futu :: Corecursive t => (a -> Base t (Free (Base t) a)) -> a -> t
futu :: forall t a.
Corecursive t =>
(a -> Base t (Free (Base t) a)) -> a -> t
futu = (forall b. Free (Base t) (Base t b) -> Base t (Free (Base t) b))
-> (a -> Base t (Free (Base t) a)) -> a -> t
forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gana forall b. Free (Base t) (Base t b) -> Base t (Free (Base t) b)
forall (f :: * -> *) a. Functor f => Free f (f a) -> f (Free f a)
distFutu

gfutu :: (Corecursive t, Functor m, Monad m) => (forall b. m (Base t b) -> Base t (m b)) -> (a -> Base t (FreeT (Base t) m a)) -> a -> t
gfutu :: forall t (m :: * -> *) a.
(Corecursive t, Functor m, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (FreeT (Base t) m a)) -> a -> t
gfutu forall b. m (Base t b) -> Base t (m b)
g = (forall b.
 FreeT (Base t) m (Base t b) -> Base t (FreeT (Base t) m b))
-> (a -> Base t (FreeT (Base t) m a)) -> a -> t
forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gana ((forall b. m (Base t b) -> Base t (m b))
-> FreeT (Base t) m (Base t b) -> Base t (FreeT (Base t) m b)
forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. h (f b) -> f (h b))
-> FreeT f h (f a) -> f (FreeT f h a)
distGFutu forall b. m (Base t b) -> Base t (m b)
g)

distFutu :: Functor f => Free f (f a) -> f (Free f a)
distFutu :: forall (f :: * -> *) a. Functor f => Free f (f a) -> f (Free f a)
distFutu (Pure f a
fx) = a -> Free f a
forall (f :: * -> *) a. a -> Free f a
Pure (a -> Free f a) -> f a -> f (Free f a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f a
fx
distFutu (Free f (Free f (f a))
ff) = f (Free f a) -> Free f a
forall (f :: * -> *) a. f (Free f a) -> Free f a
Free (f (Free f a) -> Free f a)
-> (Free f (f a) -> f (Free f a)) -> Free f (f a) -> Free f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Free f (f a) -> f (Free f a)
forall (f :: * -> *) a. Functor f => Free f (f a) -> f (Free f a)
distFutu (Free f (f a) -> Free f a) -> f (Free f (f a)) -> f (Free f a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (Free f (f a))
ff

distGFutu :: (Functor f, Functor h) => (forall b. h (f b) -> f (h b)) -> FreeT f h (f a) -> f (FreeT f h a)
distGFutu :: forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. h (f b) -> f (h b))
-> FreeT f h (f a) -> f (FreeT f h a)
distGFutu forall b. h (f b) -> f (h b)
k = FreeT f h (f a) -> f (FreeT f h a)
d where
  d :: FreeT f h (f a) -> f (FreeT f h a)
d = (h (FreeF f a (FreeT f h a)) -> FreeT f h a)
-> f (h (FreeF f a (FreeT f h a))) -> f (FreeT f h a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap h (FreeF f a (FreeT f h a)) -> FreeT f h a
forall (f :: * -> *) (m :: * -> *) a.
m (FreeF f a (FreeT f m a)) -> FreeT f m a
FreeT (f (h (FreeF f a (FreeT f h a))) -> f (FreeT f h a))
-> (FreeT f h (f a) -> f (h (FreeF f a (FreeT f h a))))
-> FreeT f h (f a)
-> f (FreeT f h a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. h (f (FreeF f a (FreeT f h a))) -> f (h (FreeF f a (FreeT f h a)))
forall b. h (f b) -> f (h b)
k (h (f (FreeF f a (FreeT f h a)))
 -> f (h (FreeF f a (FreeT f h a))))
-> (FreeT f h (f a) -> h (f (FreeF f a (FreeT f h a))))
-> FreeT f h (f a)
-> f (h (FreeF f a (FreeT f h a)))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (FreeF f (f a) (FreeT f h (f a)) -> f (FreeF f a (FreeT f h a)))
-> h (FreeF f (f a) (FreeT f h (f a)))
-> h (f (FreeF f a (FreeT f h a)))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap FreeF f (f a) (FreeT f h (f a)) -> f (FreeF f a (FreeT f h a))
d' (h (FreeF f (f a) (FreeT f h (f a)))
 -> h (f (FreeF f a (FreeT f h a))))
-> (FreeT f h (f a) -> h (FreeF f (f a) (FreeT f h (f a))))
-> FreeT f h (f a)
-> h (f (FreeF f a (FreeT f h a)))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FreeT f h (f a) -> h (FreeF f (f a) (FreeT f h (f a)))
forall (f :: * -> *) (m :: * -> *) a.
FreeT f m a -> m (FreeF f a (FreeT f m a))
runFreeT
  d' :: FreeF f (f a) (FreeT f h (f a)) -> f (FreeF f a (FreeT f h a))
d' (CMTF.Pure f a
ff) = a -> FreeF f a (FreeT f h a)
forall (f :: * -> *) a b. a -> FreeF f a b
CMTF.Pure (a -> FreeF f a (FreeT f h a))
-> f a -> f (FreeF f a (FreeT f h a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f a
ff
  d' (CMTF.Free f (FreeT f h (f a))
ff) = f (FreeT f h a) -> FreeF f a (FreeT f h a)
forall (f :: * -> *) a b. f b -> FreeF f a b
CMTF.Free (f (FreeT f h a) -> FreeF f a (FreeT f h a))
-> (FreeT f h (f a) -> f (FreeT f h a))
-> FreeT f h (f a)
-> FreeF f a (FreeT f h a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FreeT f h (f a) -> f (FreeT f h a)
d (FreeT f h (f a) -> FreeF f a (FreeT f h a))
-> f (FreeT f h (f a)) -> f (FreeF f a (FreeT f h a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (FreeT f h (f a))
ff

-------------------------------------------------------------------------------
-- Fix
-------------------------------------------------------------------------------

type instance Base (Fix f) = f
instance Functor f => Recursive (Fix f) where
  project :: Fix f -> Base (Fix f) (Fix f)
project (Fix f (Fix f)
a) = f (Fix f)
Base (Fix f) (Fix f)
a
instance Functor f => Corecursive (Fix f) where
  embed :: Base (Fix f) (Fix f) -> Fix f
embed = Base (Fix f) (Fix f) -> Fix f
forall (f :: * -> *). f (Fix f) -> Fix f
Fix

-- | Convert from one recursive type to another.
--
-- >>> showTree $ hoist (\(NonEmptyF h t) -> NodeF [h] (maybeToList t)) ( 'a' :| "bcd")
-- (a (b (c d)))
--
hoist :: (Recursive s, Corecursive t)
      => (forall a. Base s a -> Base t a) -> s -> t
hoist :: forall s t.
(Recursive s, Corecursive t) =>
(forall a. Base s a -> Base t a) -> s -> t
hoist forall a. Base s a -> Base t a
n = (Base s t -> t) -> s -> t
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata (Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (Base t t -> t) -> (Base s t -> Base t t) -> Base s t -> t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Base s t -> Base t t
forall a. Base s a -> Base t a
n)

-- | Convert from one recursive representation to another.
--
-- >>> refix ["foo", "bar"] :: Fix (ListF String)
-- Fix (Cons "foo" (Fix (Cons "bar" (Fix Nil))))
--
refix :: (Recursive s, Corecursive t, Base s ~ Base t) => s -> t
refix :: forall s t. (Recursive s, Corecursive t, Base s ~ Base t) => s -> t
refix = (Base s t -> t) -> s -> t
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata Base s t -> t
forall t. Corecursive t => Base t t -> t
embed

-------------------------------------------------------------------------------
-- Lambek
-------------------------------------------------------------------------------

-- | Lambek's lemma provides a default definition for 'project' in terms of 'cata' and 'embed'
lambek :: (Recursive t, Corecursive t) => (t -> Base t t)
lambek :: forall t. (Recursive t, Corecursive t) => t -> Base t t
lambek = (Base t (Base t t) -> Base t t) -> t -> Base t t
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata ((Base t t -> t) -> Base t (Base t t) -> Base t t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Base t t -> t
forall t. Corecursive t => Base t t -> t
embed)

-- | The dual of Lambek's lemma, provides a default definition for 'embed' in terms of 'ana' and 'project'
colambek :: (Recursive t, Corecursive t) => (Base t t -> t)
colambek :: forall t. (Recursive t, Corecursive t) => Base t t -> t
colambek = (Base t t -> Base t (Base t t)) -> Base t t -> t
forall t a. Corecursive t => (a -> Base t a) -> a -> t
ana ((t -> Base t t) -> Base t t -> Base t (Base t t)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap t -> Base t t
forall t. Recursive t => t -> Base t t
project)

type instance Base (Mu f) = f
instance Functor f => Recursive (Mu f) where
  project :: Mu f -> Base (Mu f) (Mu f)
project = Mu f -> Base (Mu f) (Mu f)
forall t. (Recursive t, Corecursive t) => t -> Base t t
lambek
  cata :: forall a. (Base (Mu f) a -> a) -> Mu f -> a
cata Base (Mu f) a -> a
f (Mu forall a. (f a -> a) -> a
g) = (f a -> a) -> a
forall a. (f a -> a) -> a
g f a -> a
Base (Mu f) a -> a
f
instance Functor f => Corecursive (Mu f) where
  embed :: Base (Mu f) (Mu f) -> Mu f
embed Base (Mu f) (Mu f)
m = (forall a. (f a -> a) -> a) -> Mu f
forall (f :: * -> *). (forall a. (f a -> a) -> a) -> Mu f
Mu (\f a -> a
f -> f a -> a
f ((Mu f -> a) -> f (Mu f) -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((Base (Mu f) a -> a) -> Mu f -> a
forall t a. Recursive t => (Base t a -> a) -> t -> a
fold f a -> a
Base (Mu f) a -> a
f) f (Mu f)
Base (Mu f) (Mu f)
m))

type instance Base (Nu f) = f
instance Functor f => Corecursive (Nu f) where
  embed :: Base (Nu f) (Nu f) -> Nu f
embed = Base (Nu f) (Nu f) -> Nu f
forall t. (Recursive t, Corecursive t) => Base t t -> t
colambek
  ana :: forall a. (a -> Base (Nu f) a) -> a -> Nu f
ana = (a -> Base (Nu f) a) -> a -> Nu f
forall (f :: * -> *) a. (a -> f a) -> a -> Nu f
Nu
instance Functor f => Recursive (Nu f) where
  project :: Nu f -> Base (Nu f) (Nu f)
project (Nu a -> f a
f a
a) = (a -> f a) -> a -> Nu f
forall (f :: * -> *) a. (a -> f a) -> a -> Nu f
Nu a -> f a
f (a -> Nu f) -> f a -> f (Nu f)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> f a
f a
a

-- | Church encoded free monads are Recursive/Corecursive, in the same way that
-- 'Mu' is.
type instance Base (CMFC.F f a) = FreeF f a
cmfcCata :: (a -> r) -> (f r -> r) -> CMFC.F f a -> r
cmfcCata :: forall a r (f :: * -> *). (a -> r) -> (f r -> r) -> F f a -> r
cmfcCata a -> r
p f r -> r
f (CMFC.F forall r. (a -> r) -> (f r -> r) -> r
run) = (a -> r) -> (f r -> r) -> r
forall r. (a -> r) -> (f r -> r) -> r
run a -> r
p f r -> r
f
instance Functor f => Recursive (CMFC.F f a) where
  project :: F f a -> Base (F f a) (F f a)
project = F f a -> Base (F f a) (F f a)
forall t. (Recursive t, Corecursive t) => t -> Base t t
lambek
  cata :: forall a. (Base (F f a) a -> a) -> F f a -> a
cata Base (F f a) a -> a
f = (a -> a) -> (f a -> a) -> F f a -> a
forall a r (f :: * -> *). (a -> r) -> (f r -> r) -> F f a -> r
cmfcCata (FreeF f a a -> a
Base (F f a) a -> a
f (FreeF f a a -> a) -> (a -> FreeF f a a) -> a -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> FreeF f a a
forall (f :: * -> *) a b. a -> FreeF f a b
CMTF.Pure) (FreeF f a a -> a
Base (F f a) a -> a
f (FreeF f a a -> a) -> (f a -> FreeF f a a) -> f a -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. f a -> FreeF f a a
forall (f :: * -> *) a b. f b -> FreeF f a b
CMTF.Free)
instance Functor f => Corecursive (CMFC.F f a) where
  embed :: Base (F f a) (F f a) -> F f a
embed (CMTF.Pure a
a)  = (forall r. (a -> r) -> (f r -> r) -> r) -> F f a
forall (f :: * -> *) a.
(forall r. (a -> r) -> (f r -> r) -> r) -> F f a
CMFC.F ((forall r. (a -> r) -> (f r -> r) -> r) -> F f a)
-> (forall r. (a -> r) -> (f r -> r) -> r) -> F f a
forall a b. (a -> b) -> a -> b
$ \a -> r
p f r -> r
_ -> a -> r
p a
a
  embed (CMTF.Free f (F f a)
fr) = (forall r. (a -> r) -> (f r -> r) -> r) -> F f a
forall (f :: * -> *) a.
(forall r. (a -> r) -> (f r -> r) -> r) -> F f a
CMFC.F ((forall r. (a -> r) -> (f r -> r) -> r) -> F f a)
-> (forall r. (a -> r) -> (f r -> r) -> r) -> F f a
forall a b. (a -> b) -> a -> b
$ \a -> r
p f r -> r
f -> f r -> r
f (f r -> r) -> f r -> r
forall a b. (a -> b) -> a -> b
$ (F f a -> r) -> f (F f a) -> f r
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((a -> r) -> (f r -> r) -> F f a -> r
forall a r (f :: * -> *). (a -> r) -> (f r -> r) -> F f a -> r
cmfcCata a -> r
p f r -> r
f) f (F f a)
fr

-- TODO: link from 'para' to 'zygo'
zygo :: Recursive t => (Base t b -> b) -> (Base t (b, a) -> a) -> t -> a
zygo :: forall t b a.
Recursive t =>
(Base t b -> b) -> (Base t (b, a) -> a) -> t -> a
zygo Base t b -> b
f = (forall b. Base t (b, b) -> (b, Base t b))
-> (Base t (b, a) -> a) -> t -> a
forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gfold ((Base t b -> b) -> Base t (b, b) -> (b, Base t b)
forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> f (b, a) -> (b, f a)
distZygo Base t b -> b
f)

distZygo
  :: Functor f
  => (f b -> b)             -- An f-algebra
  -> (f (b, a) -> (b, f a)) -- ^ A distributive for semi-mutual recursion
distZygo :: forall (f :: * -> *) b a.
Functor f =>
(f b -> b) -> f (b, a) -> (b, f a)
distZygo f b -> b
g f (b, a)
m = (f b -> b
g (((b, a) -> b) -> f (b, a) -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (b, a) -> b
forall a b. (a, b) -> a
fst f (b, a)
m), ((b, a) -> a) -> f (b, a) -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (b, a) -> a
forall a b. (a, b) -> b
snd f (b, a)
m)

gzygo
  :: (Recursive t, Comonad w)
  => (Base t b -> b)
  -> (forall c. Base t (w c) -> w (Base t c))
  -> (Base t (EnvT b w a) -> a)
  -> t
  -> a
gzygo :: forall t (w :: * -> *) b a.
(Recursive t, Comonad w) =>
(Base t b -> b)
-> (forall c. Base t (w c) -> w (Base t c))
-> (Base t (EnvT b w a) -> a)
-> t
-> a
gzygo Base t b -> b
f forall c. Base t (w c) -> w (Base t c)
w = (forall b. Base t (EnvT b w b) -> EnvT b w (Base t b))
-> (Base t (EnvT b w a) -> a) -> t -> a
forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gfold ((Base t b -> b)
-> (forall c. Base t (w c) -> w (Base t c))
-> Base t (EnvT b w b)
-> EnvT b w (Base t b)
forall (f :: * -> *) (w :: * -> *) b a.
(Functor f, Comonad w) =>
(f b -> b)
-> (forall c. f (w c) -> w (f c))
-> f (EnvT b w a)
-> EnvT b w (f a)
distZygoT Base t b -> b
f forall c. Base t (w c) -> w (Base t c)
w)

distZygoT
  :: (Functor f, Comonad w)
  => (f b -> b)                        -- An f-w-algebra to use for semi-mutual recursion
  -> (forall c. f (w c) -> w (f c))    -- A base Distributive law
  -> f (EnvT b w a) -> EnvT b w (f a)  -- A new distributive law that adds semi-mutual recursion
distZygoT :: forall (f :: * -> *) (w :: * -> *) b a.
(Functor f, Comonad w) =>
(f b -> b)
-> (forall c. f (w c) -> w (f c))
-> f (EnvT b w a)
-> EnvT b w (f a)
distZygoT f b -> b
g forall c. f (w c) -> w (f c)
k f (EnvT b w a)
fe = b -> w (f a) -> EnvT b w (f a)
forall e (w :: * -> *) a. e -> w a -> EnvT e w a
EnvT (f b -> b
g (EnvT b w a -> b
forall {e} {w :: * -> *} {a}. EnvT e w a -> e
getEnv (EnvT b w a -> b) -> f (EnvT b w a) -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (EnvT b w a)
fe)) (f (w a) -> w (f a)
forall c. f (w c) -> w (f c)
k (EnvT b w a -> w a
forall (t :: (* -> *) -> * -> *) (w :: * -> *) a.
(ComonadTrans t, Comonad w) =>
t w a -> w a
lower (EnvT b w a -> w a) -> f (EnvT b w a) -> f (w a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f (EnvT b w a)
fe))
  where getEnv :: EnvT e w a -> e
getEnv (EnvT e
e w a
_) = e
e

gapo :: Corecursive t => (b -> Base t b) -> (a -> Base t (Either b a)) -> a -> t
gapo :: forall t b a.
Corecursive t =>
(b -> Base t b) -> (a -> Base t (Either b a)) -> a -> t
gapo b -> Base t b
g = (forall b. Either b (Base t b) -> Base t (Either b b))
-> (a -> Base t (Either b a)) -> a -> t
forall t (m :: * -> *) a.
(Corecursive t, Monad m) =>
(forall b. m (Base t b) -> Base t (m b))
-> (a -> Base t (m a)) -> a -> t
gunfold ((b -> Base t b) -> Either b (Base t b) -> Base t (Either b b)
forall (f :: * -> *) b a.
Functor f =>
(b -> f b) -> Either b (f a) -> f (Either b a)
distGApo b -> Base t b
g)

distApo :: Recursive t => Either t (Base t a) -> Base t (Either t a)
distApo :: forall t a.
Recursive t =>
Either t (Base t a) -> Base t (Either t a)
distApo = (t -> Base t t) -> Either t (Base t a) -> Base t (Either t a)
forall (f :: * -> *) b a.
Functor f =>
(b -> f b) -> Either b (f a) -> f (Either b a)
distGApo t -> Base t t
forall t. Recursive t => t -> Base t t
project

distGApo :: Functor f => (b -> f b) -> Either b (f a) -> f (Either b a)
distGApo :: forall (f :: * -> *) b a.
Functor f =>
(b -> f b) -> Either b (f a) -> f (Either b a)
distGApo b -> f b
f = (b -> f (Either b a))
-> (f a -> f (Either b a)) -> Either b (f a) -> f (Either b a)
forall a c b. (a -> c) -> (b -> c) -> Either a b -> c
either ((b -> Either b a) -> f b -> f (Either b a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap b -> Either b a
forall a b. a -> Either a b
Left (f b -> f (Either b a)) -> (b -> f b) -> b -> f (Either b a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. b -> f b
f) ((a -> Either b a) -> f a -> f (Either b a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> Either b a
forall a b. b -> Either a b
Right)

distGApoT
  :: (Functor f, Functor m)
  => (b -> f b)
  -> (forall c. m (f c) -> f (m c))
  -> ExceptT b m (f a)
  -> f (ExceptT b m a)
distGApoT :: forall (f :: * -> *) (m :: * -> *) b a.
(Functor f, Functor m) =>
(b -> f b)
-> (forall c. m (f c) -> f (m c))
-> ExceptT b m (f a)
-> f (ExceptT b m a)
distGApoT b -> f b
g forall c. m (f c) -> f (m c)
k = (m (Either b a) -> ExceptT b m a)
-> f (m (Either b a)) -> f (ExceptT b m a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap m (Either b a) -> ExceptT b m a
forall e (m :: * -> *) a. m (Either e a) -> ExceptT e m a
ExceptT (f (m (Either b a)) -> f (ExceptT b m a))
-> (ExceptT b m (f a) -> f (m (Either b a)))
-> ExceptT b m (f a)
-> f (ExceptT b m a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. m (f (Either b a)) -> f (m (Either b a))
forall c. m (f c) -> f (m c)
k (m (f (Either b a)) -> f (m (Either b a)))
-> (ExceptT b m (f a) -> m (f (Either b a)))
-> ExceptT b m (f a)
-> f (m (Either b a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Either b (f a) -> f (Either b a))
-> m (Either b (f a)) -> m (f (Either b a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((b -> f b) -> Either b (f a) -> f (Either b a)
forall (f :: * -> *) b a.
Functor f =>
(b -> f b) -> Either b (f a) -> f (Either b a)
distGApo b -> f b
g) (m (Either b (f a)) -> m (f (Either b a)))
-> (ExceptT b m (f a) -> m (Either b (f a)))
-> ExceptT b m (f a)
-> m (f (Either b a))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ExceptT b m (f a) -> m (Either b (f a))
forall e (m :: * -> *) a. ExceptT e m a -> m (Either e a)
runExceptT

-- | Course-of-value iteration
histo :: Recursive t => (Base t (Cofree (Base t) a) -> a) -> t -> a
histo :: forall t a.
Recursive t =>
(Base t (Cofree (Base t) a) -> a) -> t -> a
histo = (forall b.
 Base t (Cofree (Base t) b) -> Cofree (Base t) (Base t b))
-> (Base t (Cofree (Base t) a) -> a) -> t -> a
forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gcata forall b. Base t (Cofree (Base t) b) -> Cofree (Base t) (Base t b)
forall (f :: * -> *) a.
Functor f =>
f (Cofree f a) -> Cofree f (f a)
distHisto

ghisto :: (Recursive t, Comonad w) => (forall b. Base t (w b) -> w (Base t b)) -> (Base t (CofreeT (Base t) w a) -> a) -> t -> a
ghisto :: forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (CofreeT (Base t) w a) -> a) -> t -> a
ghisto forall b. Base t (w b) -> w (Base t b)
g = (forall b.
 Base t (CofreeT (Base t) w b) -> CofreeT (Base t) w (Base t b))
-> (Base t (CofreeT (Base t) w a) -> a) -> t -> a
forall t (w :: * -> *) a.
(Recursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (Base t (w a) -> a) -> t -> a
gcata ((forall b. Base t (w b) -> w (Base t b))
-> Base t (CofreeT (Base t) w b) -> CofreeT (Base t) w (Base t b)
forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. f (h b) -> h (f b))
-> f (CofreeT f h a) -> CofreeT f h (f a)
distGHisto forall b. Base t (w b) -> w (Base t b)
g)

distHisto :: Functor f => f (Cofree f a) -> Cofree f (f a)
distHisto :: forall (f :: * -> *) a.
Functor f =>
f (Cofree f a) -> Cofree f (f a)
distHisto f (Cofree f a)
fc = (Cofree f a -> a) -> f (Cofree f a) -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Cofree f a -> a
forall (w :: * -> *) a. Comonad w => w a -> a
extract f (Cofree f a)
fc f a -> f (Cofree f (f a)) -> Cofree f (f a)
forall (f :: * -> *) a. a -> f (Cofree f a) -> Cofree f a
:< (Cofree f a -> Cofree f (f a))
-> f (Cofree f a) -> f (Cofree f (f a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (f (Cofree f a) -> Cofree f (f a)
forall (f :: * -> *) a.
Functor f =>
f (Cofree f a) -> Cofree f (f a)
distHisto (f (Cofree f a) -> Cofree f (f a))
-> (Cofree f a -> f (Cofree f a)) -> Cofree f a -> Cofree f (f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Cofree f a -> f (Cofree f a)
forall (f :: * -> *) (w :: * -> *) a.
ComonadCofree f w =>
w a -> f (w a)
Cofree.unwrap) f (Cofree f a)
fc

distGHisto :: (Functor f, Functor h) => (forall b. f (h b) -> h (f b)) -> f (CofreeT f h a) -> CofreeT f h (f a)
distGHisto :: forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. f (h b) -> h (f b))
-> f (CofreeT f h a) -> CofreeT f h (f a)
distGHisto forall b. f (h b) -> h (f b)
k = f (CofreeT f h a) -> CofreeT f h (f a)
d where d :: f (CofreeT f h a) -> CofreeT f h (f a)
d = h (CofreeF f (f a) (CofreeT f h (f a))) -> CofreeT f h (f a)
forall (f :: * -> *) (w :: * -> *) a.
w (CofreeF f a (CofreeT f w a)) -> CofreeT f w a
CofreeT (h (CofreeF f (f a) (CofreeT f h (f a))) -> CofreeT f h (f a))
-> (f (CofreeT f h a) -> h (CofreeF f (f a) (CofreeT f h (f a))))
-> f (CofreeT f h a)
-> CofreeT f h (f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (f (CofreeF f a (CofreeT f h a))
 -> CofreeF f (f a) (CofreeT f h (f a)))
-> h (f (CofreeF f a (CofreeT f h a)))
-> h (CofreeF f (f a) (CofreeT f h (f a)))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (\f (CofreeF f a (CofreeT f h a))
fc -> (CofreeF f a (CofreeT f h a) -> a)
-> f (CofreeF f a (CofreeT f h a)) -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap CofreeF f a (CofreeT f h a) -> a
forall (f :: * -> *) a b. CofreeF f a b -> a
CCTC.headF f (CofreeF f a (CofreeT f h a))
fc f a -> f (CofreeT f h (f a)) -> CofreeF f (f a) (CofreeT f h (f a))
forall (f :: * -> *) a b. a -> f b -> CofreeF f a b
CCTC.:< (CofreeF f a (CofreeT f h a) -> CofreeT f h (f a))
-> f (CofreeF f a (CofreeT f h a)) -> f (CofreeT f h (f a))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (f (CofreeT f h a) -> CofreeT f h (f a)
d (f (CofreeT f h a) -> CofreeT f h (f a))
-> (CofreeF f a (CofreeT f h a) -> f (CofreeT f h a))
-> CofreeF f a (CofreeT f h a)
-> CofreeT f h (f a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. CofreeF f a (CofreeT f h a) -> f (CofreeT f h a)
forall (f :: * -> *) a b. CofreeF f a b -> f b
CCTC.tailF) f (CofreeF f a (CofreeT f h a))
fc) (h (f (CofreeF f a (CofreeT f h a)))
 -> h (CofreeF f (f a) (CofreeT f h (f a))))
-> (f (CofreeT f h a) -> h (f (CofreeF f a (CofreeT f h a))))
-> f (CofreeT f h a)
-> h (CofreeF f (f a) (CofreeT f h (f a)))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. f (h (CofreeF f a (CofreeT f h a)))
-> h (f (CofreeF f a (CofreeT f h a)))
forall b. f (h b) -> h (f b)
k (f (h (CofreeF f a (CofreeT f h a)))
 -> h (f (CofreeF f a (CofreeT f h a))))
-> (f (CofreeT f h a) -> f (h (CofreeF f a (CofreeT f h a))))
-> f (CofreeT f h a)
-> h (f (CofreeF f a (CofreeT f h a)))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (CofreeT f h a -> h (CofreeF f a (CofreeT f h a)))
-> f (CofreeT f h a) -> f (h (CofreeF f a (CofreeT f h a)))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap CofreeT f h a -> h (CofreeF f a (CofreeT f h a))
forall (f :: * -> *) (w :: * -> *) a.
CofreeT f w a -> w (CofreeF f a (CofreeT f w a))
runCofreeT

chrono :: Functor f => (f (Cofree f b) -> b) -> (a -> f (Free f a)) -> (a -> b)
chrono :: forall (f :: * -> *) b a.
Functor f =>
(f (Cofree f b) -> b) -> (a -> f (Free f a)) -> a -> b
chrono = (forall c. f (Cofree f c) -> Cofree f (f c))
-> (forall d. Free f (f d) -> f (Free f d))
-> (f (Cofree f b) -> b)
-> (a -> f (Free f a))
-> a
-> b
forall (w :: * -> *) (f :: * -> *) (m :: * -> *) b a.
(Comonad w, Functor f, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
ghylo forall c. f (Cofree f c) -> Cofree f (f c)
forall (f :: * -> *) a.
Functor f =>
f (Cofree f a) -> Cofree f (f a)
distHisto forall d. Free f (f d) -> f (Free f d)
forall (f :: * -> *) a. Functor f => Free f (f a) -> f (Free f a)
distFutu

gchrono :: (Functor f, Functor w, Functor m, Comonad w, Monad m) =>
           (forall c. f (w c) -> w (f c)) ->
           (forall c. m (f c) -> f (m c)) ->
           (f (CofreeT f w b) -> b) -> (a -> f (FreeT f m a)) ->
           (a -> b)
gchrono :: forall (f :: * -> *) (w :: * -> *) (m :: * -> *) b a.
(Functor f, Functor w, Functor m, Comonad w, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall c. m (f c) -> f (m c))
-> (f (CofreeT f w b) -> b)
-> (a -> f (FreeT f m a))
-> a
-> b
gchrono forall c. f (w c) -> w (f c)
w forall c. m (f c) -> f (m c)
m = (forall c. f (CofreeT f w c) -> CofreeT f w (f c))
-> (forall d. FreeT f m (f d) -> f (FreeT f m d))
-> (f (CofreeT f w b) -> b)
-> (a -> f (FreeT f m a))
-> a
-> b
forall (w :: * -> *) (f :: * -> *) (m :: * -> *) b a.
(Comonad w, Functor f, Monad m) =>
(forall c. f (w c) -> w (f c))
-> (forall d. m (f d) -> f (m d))
-> (f (w b) -> b)
-> (a -> f (m a))
-> a
-> b
ghylo ((forall c. f (w c) -> w (f c))
-> f (CofreeT f w c) -> CofreeT f w (f c)
forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. f (h b) -> h (f b))
-> f (CofreeT f h a) -> CofreeT f h (f a)
distGHisto forall c. f (w c) -> w (f c)
w) ((forall c. m (f c) -> f (m c))
-> FreeT f m (f d) -> f (FreeT f m d)
forall (f :: * -> *) (h :: * -> *) a.
(Functor f, Functor h) =>
(forall b. h (f b) -> f (h b))
-> FreeT f h (f a) -> f (FreeT f h a)
distGFutu forall c. m (f c) -> f (m c)
m)

-- | Mendler-style iteration
mcata :: (forall y. (y -> c) -> f y -> c) -> Fix f -> c
mcata :: forall c (f :: * -> *).
(forall y. (y -> c) -> f y -> c) -> Fix f -> c
mcata forall y. (y -> c) -> f y -> c
psi = Fix f -> c
c where c :: Fix f -> c
c = (Fix f -> c) -> f (Fix f) -> c
forall y. (y -> c) -> f y -> c
psi Fix f -> c
c (f (Fix f) -> c) -> (Fix f -> f (Fix f)) -> Fix f -> c
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Fix f -> f (Fix f)
forall (f :: * -> *). Fix f -> f (Fix f)
unFix

-- | Mendler-style recursion
--
-- @since 5.2.2
mpara :: (forall y. (y -> c) -> (y -> Fix f) -> f y -> c) -> Fix f -> c
mpara :: forall c (f :: * -> *).
(forall y. (y -> c) -> (y -> Fix f) -> f y -> c) -> Fix f -> c
mpara forall y. (y -> c) -> (y -> Fix f) -> f y -> c
psi = Fix f -> c
c where c :: Fix f -> c
c = (Fix f -> c) -> (Fix f -> Fix f) -> f (Fix f) -> c
forall y. (y -> c) -> (y -> Fix f) -> f y -> c
psi Fix f -> c
c Fix f -> Fix f
forall a. a -> a
id (f (Fix f) -> c) -> (Fix f -> f (Fix f)) -> Fix f -> c
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Fix f -> f (Fix f)
forall (f :: * -> *). Fix f -> f (Fix f)
unFix

-- | Mendler-style semi-mutual recursion
--
-- @since 5.2.2
mzygo :: (forall y. (y -> b) -> f y -> b) -> (forall y. (y -> c) -> (y -> b) -> f y -> c) -> Fix f -> c
mzygo :: forall b (f :: * -> *) c.
(forall y. (y -> b) -> f y -> b)
-> (forall y. (y -> c) -> (y -> b) -> f y -> c) -> Fix f -> c
mzygo forall y. (y -> b) -> f y -> b
phi forall y. (y -> c) -> (y -> b) -> f y -> c
psi = Fix f -> c
c where c :: Fix f -> c
c = (Fix f -> c) -> (Fix f -> b) -> f (Fix f) -> c
forall y. (y -> c) -> (y -> b) -> f y -> c
psi Fix f -> c
c ((forall y. (y -> b) -> f y -> b) -> Fix f -> b
forall c (f :: * -> *).
(forall y. (y -> c) -> f y -> c) -> Fix f -> c
mcata forall y. (y -> b) -> f y -> b
phi) (f (Fix f) -> c) -> (Fix f -> f (Fix f)) -> Fix f -> c
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Fix f -> f (Fix f)
forall (f :: * -> *). Fix f -> f (Fix f)
unFix

-- | Mendler-style course-of-value iteration
mhisto :: (forall y. (y -> c) -> (y -> f y) -> f y -> c) -> Fix f -> c
mhisto :: forall c (f :: * -> *).
(forall y. (y -> c) -> (y -> f y) -> f y -> c) -> Fix f -> c
mhisto forall y. (y -> c) -> (y -> f y) -> f y -> c
psi = Fix f -> c
c where c :: Fix f -> c
c = (Fix f -> c) -> (Fix f -> f (Fix f)) -> f (Fix f) -> c
forall y. (y -> c) -> (y -> f y) -> f y -> c
psi Fix f -> c
c Fix f -> f (Fix f)
forall (f :: * -> *). Fix f -> f (Fix f)
unFix (f (Fix f) -> c) -> (Fix f -> f (Fix f)) -> Fix f -> c
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Fix f -> f (Fix f)
forall (f :: * -> *). Fix f -> f (Fix f)
unFix

-- | Mendler-style coiteration
--
-- @since 5.2.2
mana :: (forall y. (x -> y) -> x -> f y) -> x -> Fix f
mana :: forall x (f :: * -> *).
(forall y. (x -> y) -> x -> f y) -> x -> Fix f
mana forall y. (x -> y) -> x -> f y
phi = x -> Fix f
c where c :: x -> Fix f
c = f (Fix f) -> Fix f
forall (f :: * -> *). f (Fix f) -> Fix f
Fix (f (Fix f) -> Fix f) -> (x -> f (Fix f)) -> x -> Fix f
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (x -> Fix f) -> x -> f (Fix f)
forall y. (x -> y) -> x -> f y
phi x -> Fix f
c

-- | Mendler-style corecursion
--
-- @since 5.2.2
mapo :: (forall y. (Fix f -> y) -> (x -> y) -> x -> f y) -> x -> Fix f
mapo :: forall (f :: * -> *) x.
(forall y. (Fix f -> y) -> (x -> y) -> x -> f y) -> x -> Fix f
mapo forall y. (Fix f -> y) -> (x -> y) -> x -> f y
phi = x -> Fix f
c where c :: x -> Fix f
c = f (Fix f) -> Fix f
forall (f :: * -> *). f (Fix f) -> Fix f
Fix (f (Fix f) -> Fix f) -> (x -> f (Fix f)) -> x -> Fix f
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Fix f -> Fix f) -> (x -> Fix f) -> x -> f (Fix f)
forall y. (Fix f -> y) -> (x -> y) -> x -> f y
phi Fix f -> Fix f
forall a. a -> a
id x -> Fix f
c

-- | Mendler-style course-of-values coiteration
--
-- @since 5.2.2
mfutu :: (forall y. (f y -> y) -> (x -> y) -> x -> f y) -> x -> Fix f
mfutu :: forall (f :: * -> *) x.
(forall y. (f y -> y) -> (x -> y) -> x -> f y) -> x -> Fix f
mfutu forall y. (f y -> y) -> (x -> y) -> x -> f y
phi = x -> Fix f
c where c :: x -> Fix f
c = f (Fix f) -> Fix f
forall (f :: * -> *). f (Fix f) -> Fix f
Fix (f (Fix f) -> Fix f) -> (x -> f (Fix f)) -> x -> Fix f
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (f (Fix f) -> Fix f) -> (x -> Fix f) -> x -> f (Fix f)
forall y. (f y -> y) -> (x -> y) -> x -> f y
phi f (Fix f) -> Fix f
forall (f :: * -> *). f (Fix f) -> Fix f
Fix x -> Fix f
c

-- | Elgot algebras
elgot :: Functor f => (f a -> a) -> (b -> Either a (f b)) -> b -> a
elgot :: forall (f :: * -> *) a b.
Functor f =>
(f a -> a) -> (b -> Either a (f b)) -> b -> a
elgot f a -> a
phi b -> Either a (f b)
psi = b -> a
h where h :: b -> a
h = (a -> a
forall a. a -> a
id (a -> a) -> (f b -> a) -> Either a (f b) -> a
forall (a :: * -> * -> *) b d c.
ArrowChoice a =>
a b d -> a c d -> a (Either b c) d
||| f a -> a
phi (f a -> a) -> (f b -> f a) -> f b -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (b -> a) -> f b -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap b -> a
h) (Either a (f b) -> a) -> (b -> Either a (f b)) -> b -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. b -> Either a (f b)
psi

-- | Elgot coalgebras: <http://comonad.com/reader/2008/elgot-coalgebras/>
coelgot :: Functor f => ((a, f b) -> b) -> (a -> f a) -> a -> b
coelgot :: forall (f :: * -> *) a b.
Functor f =>
((a, f b) -> b) -> (a -> f a) -> a -> b
coelgot (a, f b) -> b
phi a -> f a
psi = a -> b
h where h :: a -> b
h = (a, f b) -> b
phi ((a, f b) -> b) -> (a -> (a, f b)) -> a -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> a
forall a. a -> a
id (a -> a) -> (a -> f b) -> a -> (a, f b)
forall (a :: * -> * -> *) b c c'.
Arrow a =>
a b c -> a b c' -> a b (c, c')
&&& (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
h (f a -> f b) -> (a -> f a) -> a -> f b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> f a
psi)

-- | Zygohistomorphic prepromorphisms:
--
-- A corrected and modernized version of <http://www.haskell.org/haskellwiki/Zygohistomorphic_prepromorphisms>
zygoHistoPrepro
  :: (Corecursive t, Recursive t)
  => (Base t b -> b)
  -> (forall c. Base t c -> Base t c)
  -> (Base t (EnvT b (Cofree (Base t)) a) -> a)
  -> t
  -> a
zygoHistoPrepro :: forall t b a.
(Corecursive t, Recursive t) =>
(Base t b -> b)
-> (forall c. Base t c -> Base t c)
-> (Base t (EnvT b (Cofree (Base t)) a) -> a)
-> t
-> a
zygoHistoPrepro Base t b -> b
f forall c. Base t c -> Base t c
g Base t (EnvT b (Cofree (Base t)) a) -> a
t = (forall b.
 Base t (EnvT b (Cofree (Base t)) b)
 -> EnvT b (Cofree (Base t)) (Base t b))
-> (forall c. Base t c -> Base t c)
-> (Base t (EnvT b (Cofree (Base t)) a) -> a)
-> t
-> a
forall t (w :: * -> *) a.
(Recursive t, Corecursive t, Comonad w) =>
(forall b. Base t (w b) -> w (Base t b))
-> (forall b. Base t b -> Base t b)
-> (Base t (w a) -> a)
-> t
-> a
gprepro ((Base t b -> b)
-> (forall c.
    Base t (Cofree (Base t) c) -> Cofree (Base t) (Base t c))
-> Base t (EnvT b (Cofree (Base t)) b)
-> EnvT b (Cofree (Base t)) (Base t b)
forall (f :: * -> *) (w :: * -> *) b a.
(Functor f, Comonad w) =>
(f b -> b)
-> (forall c. f (w c) -> w (f c))
-> f (EnvT b w a)
-> EnvT b w (f a)
distZygoT Base t b -> b
f forall c. Base t (Cofree (Base t) c) -> Cofree (Base t) (Base t c)
forall (f :: * -> *) a.
Functor f =>
f (Cofree f a) -> Cofree f (f a)
distHisto) forall c. Base t c -> Base t c
g Base t (EnvT b (Cofree (Base t)) a) -> a
t

-------------------------------------------------------------------------------
-- Effectful combinators
-------------------------------------------------------------------------------

-- | A specialization of 'cata' for effectful folds.
--
-- 'cataA' is the same as 'cata', but with a more specialized type. The only
-- reason it exists is to make it easier to discover how to use this library
-- with effects.
--
-- For our running example, let's improve the output format of our
-- pretty-printer by using indentation. To do so, we will need to keep track of
-- the current indentation level. We will do so using a @Reader Int@ effect.
-- Our recursive positions will thus contain @Reader Int String@ actions, not
-- @String@s. This means we need to run those actions in order to get the
-- results.
--
-- >>> :{
-- let pprint2 :: Tree Int -> String
--     pprint2 = flip runReader 0 . cataA go
--       where
--         go :: TreeF Int (Reader Int String)
--            -> Reader Int String
--         go (NodeF i rss) = do
--           -- rss :: [Reader Int String]
--           -- ss  :: [String]
--           ss <- local (+ 2) $ sequence rss
--           indent <- ask
--           let s = replicate indent ' ' ++ "* " ++ show i
--           pure $ intercalate "\n" (s : ss)
-- :}
--
-- >>> putStrLn $ pprint2 myTree
-- * 0
--   * 1
--   * 2
--   * 3
--     * 31
--       * 311
--         * 3111
--         * 3112
--
-- The fact that the recursive positions contain 'Reader' actions instead of
-- 'String's gives us some flexibility. Here, we are able to increase the
-- indentation by running those actions inside a 'local' block. More generally,
-- we can control the order of their side-effects, interleave them with other
-- effects, etc.
--
-- A similar technique is to specialize 'cata' so that the result is a
-- function. This makes it possible for data to flow down in addition to up.
-- In this modified version of our running example, the indentation level flows
-- down from the root to the leaves, while the resulting strings flow up from
-- the leaves to the root.
--
-- >>> :{
-- let pprint3 :: Tree Int -> String
--     pprint3 t = cataA go t 0
--       where
--         go :: TreeF Int (Int -> String)
--            -> Int -> String
--         go (NodeF i fs) indent =
--           -- fs :: [Int -> String]
--           let indent' = indent + 2
--               ss = map (\f -> f indent') fs
--               s = replicate indent ' ' ++ "* " ++ show i
--           in intercalate "\n" (s : ss)
-- :}
--
-- >>> putStrLn $ pprint3 myTree
-- * 0
--   * 1
--   * 2
--   * 3
--     * 31
--       * 311
--         * 3111
--         * 3112
cataA :: (Recursive t) => (Base t (f a) -> f a) -> t -> f a
cataA :: forall t (f :: * -> *) a.
Recursive t =>
(Base t (f a) -> f a) -> t -> f a
cataA = (Base t (f a) -> f a) -> t -> f a
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata

-- | An effectful version of 'hoist'.
--
-- Properties:
--
-- @
-- 'transverse' 'sequenceA' = 'pure'
-- @
--
-- Examples:
--
-- The weird type of first argument allows user to decide
-- an order of sequencing:
--
-- >>> transverse (\x -> print (void x) *> sequence x) "foo" :: IO String
-- Cons 'f' ()
-- Cons 'o' ()
-- Cons 'o' ()
-- Nil
-- "foo"
--
-- >>> transverse (\x -> sequence x <* print (void x)) "foo" :: IO String
-- Nil
-- Cons 'o' ()
-- Cons 'o' ()
-- Cons 'f' ()
-- "foo"
--
transverse :: (Recursive s, Corecursive t, Functor f)
           => (forall a. Base s (f a) -> f (Base t a)) -> s -> f t
transverse :: forall s t (f :: * -> *).
(Recursive s, Corecursive t, Functor f) =>
(forall a. Base s (f a) -> f (Base t a)) -> s -> f t
transverse forall a. Base s (f a) -> f (Base t a)
n = (Base s (f t) -> f t) -> s -> f t
forall t a. Recursive t => (Base t a -> a) -> t -> a
cata ((Base t t -> t) -> f (Base t t) -> f t
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Base t t -> t
forall t. Corecursive t => Base t t -> t
embed (f (Base t t) -> f t)
-> (Base s (f t) -> f (Base t t)) -> Base s (f t) -> f t
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Base s (f t) -> f (Base t t)
forall a. Base s (f a) -> f (Base t a)
n)

-- | A coeffectful version of 'hoist'.
--
-- Properties:
--
-- @
-- 'cotransverse' 'distAna' = 'runIdentity'
-- @
--
-- Examples:
--
-- Stateful transformations:
--
-- >>> :{
-- cotransverse
--   (\(u, b) -> case b of
--     Nil -> Nil
--     Cons x a -> Cons (if u then toUpper x else x) (not u, a))
--   (True, "foobar") :: String
-- :}
-- "FoObAr"
--
-- We can implement a variant of `zipWith`
--
-- >>> data Pair a = Pair a a deriving Functor
--
-- >>> :{
-- let zipWith' :: forall a b. (a -> a -> b) -> [a] -> [a] -> [b]
--     zipWith' f xs ys = cotransverse g (Pair xs ys) where
--       g :: Pair (ListF a c) -> ListF b (Pair c)
--       g (Pair Nil        _)          = Nil
--       g (Pair _          Nil)        = Nil
--       g (Pair (Cons x a) (Cons y b)) = Cons (f x y) (Pair a b)
--     :}
--
-- >>> zipWith' (*) [1,2,3] [4,5,6]
-- [4,10,18]
--
-- >>> zipWith' (*) [1,2,3] [4,5,6,8]
-- [4,10,18]
--
-- >>> zipWith' (*) [1,2,3,3] [4,5,6]
-- [4,10,18]
--
cotransverse :: (Recursive s, Corecursive t, Functor f)
             => (forall a. f (Base s a) -> Base t (f a)) -> f s -> t
cotransverse :: forall s t (f :: * -> *).
(Recursive s, Corecursive t, Functor f) =>
(forall a. f (Base s a) -> Base t (f a)) -> f s -> t
cotransverse forall a. f (Base s a) -> Base t (f a)
n = (f s -> Base t (f s)) -> f s -> t
forall t a. Corecursive t => (a -> Base t a) -> a -> t
ana (f (Base s s) -> Base t (f s)
forall a. f (Base s a) -> Base t (f a)
n (f (Base s s) -> Base t (f s))
-> (f s -> f (Base s s)) -> f s -> Base t (f s)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (s -> Base s s) -> f s -> f (Base s s)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap s -> Base s s
forall t. Recursive t => t -> Base t t
project)

-------------------------------------------------------------------------------
-- GCoerce
-------------------------------------------------------------------------------

class GCoerce f g where
    gcoerce :: f a -> g a

instance GCoerce f g => GCoerce (M1 i c f) (M1 i c' g) where
    gcoerce :: forall a. M1 i c f a -> M1 i c' g a
gcoerce (M1 f a
x) = g a -> M1 i c' g a
forall k i (c :: Meta) (f :: k -> *) (p :: k). f p -> M1 i c f p
M1 (f a -> g a
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce f a
x)

-- R changes to/from P with GHC-7.4.2 at least.
instance GCoerce (K1 i c) (K1 j c) where
    gcoerce :: forall a. K1 i c a -> K1 j c a
gcoerce = c -> K1 j c a
forall k i c (p :: k). c -> K1 i c p
K1 (c -> K1 j c a) -> (K1 i c a -> c) -> K1 i c a -> K1 j c a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. K1 i c a -> c
forall k i c (p :: k). K1 i c p -> c
unK1

instance GCoerce U1 U1 where
    gcoerce :: forall a. U1 a -> U1 a
gcoerce = U1 a -> U1 a
forall a. a -> a
id

instance GCoerce V1 V1 where
    gcoerce :: forall a. V1 a -> V1 a
gcoerce = V1 a -> V1 a
forall a. a -> a
id

instance (GCoerce f g, GCoerce f' g') => GCoerce (f :*: f') (g :*: g') where
    gcoerce :: forall a. (:*:) f f' a -> (:*:) g g' a
gcoerce (f a
x :*: f' a
y) = f a -> g a
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce f a
x g a -> g' a -> (:*:) g g' a
forall k (f :: k -> *) (g :: k -> *) (p :: k).
f p -> g p -> (:*:) f g p
:*: f' a -> g' a
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce f' a
y

instance (GCoerce f g, GCoerce f' g') => GCoerce (f :+: f') (g :+: g') where
    gcoerce :: forall a. (:+:) f f' a -> (:+:) g g' a
gcoerce (L1 f a
x) = g a -> (:+:) g g' a
forall k (f :: k -> *) (g :: k -> *) (p :: k). f p -> (:+:) f g p
L1 (f a -> g a
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce f a
x)
    gcoerce (R1 f' a
x) = g' a -> (:+:) g g' a
forall k (f :: k -> *) (g :: k -> *) (p :: k). g p -> (:+:) f g p
R1 (f' a -> g' a
forall (f :: * -> *) (g :: * -> *) a. GCoerce f g => f a -> g a
gcoerce f' a
x)