# deepcontrol

A Haskell library that enables more deeper level style programming than the usual Control.xxx modules provide, especially for Applicative and Monad.

## Examples

### [Applicative](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Applicative.html)

This module enables you to program in applicative style for more deeper level than the usual Applicative module expresses.
You would soon realize exactly what more deeper level means by reading the example codes below in order.

    Prelude> :m DeepControl.Applicative

#### Level-0

bra-ket notation:

    > (1+) |> 2
    3
    > 1 <| (+2)
    3

    > 1 <|(+)|> 2
    3
    > 1 <|(+)|> 2 <|(*)|> 3
    9

    > 1 <|(,)|> 2
    (1,2)

#### Level-1

bra-ket notation:

    > (1+) |$> [2]
    [3]
    > [1] <$| (+2)
    [3]
    > ("<"++)|$> ["a","b"] <$|(++">")
    ["<a>","<b>"]

    > [(1+)] |*> [2]
    [3]

    > [1] <$|(+)|*> [2]
    [3]
    > [1] <$|(+)|*> [0,1,2]
    [1,2,3]
    > [0,1] <$|(+)|*> [2,3] <$|(+)|*> [4,5]
    [6,7,7,8,7,8,8,9]

    > foldr (\x acc -> x <$|(:)|*> acc) ((.*) []) [Just 1, Just 2,  Just 3]
    Just [1,2,3]
    > foldr (\x acc -> x <$|(:)|*> acc) ((.*) []) [Just 1, Nothing, Just 3]
    Nothing

    > filter (even <$|(&&)|*> (10 >)) [1..100]
    [2,4,6,8]
    > filter (even <$|(&&)|*> (10 >) <$|(&&)|*> (5 <)) [1..100]
    [6,8]

cover notation:

    > :t (.*)
    (.*) :: Applicative f => a -> f a

    > (.*) 1 :: Maybe Int
    Just 1
    > (.*) 1 :: [Int]
    [1]
    > (.*) 1 :: Either () Int
    Right 1

cover-braket notation:

    > :t (|*)
    (|*) :: Applicative f => f (a -> b) -> a -> f b

    > [(1+)] |* 2
    [3]
    > [1] <$|(+)|* 2
    [3]

    > (,) |$> ["a1","a2"] |* 'b'
    [("a1",'b'),("a2",'b')]

    > (,,) 'a' |$> ["b1","b2"] |* 'c'
    [('a',"b1",'c'),('a',"b2",'c')]

    > (,,,) 'a' |$> ["b1","b2"] |* 'c' |* 'd'
    [('a',"b1",'c','d'),('a',"b2",'c','d')]
    > (,,,) 'a' |$> ["b1","b2"] |* 'c' |*> ["d1","d2"]
    [('a',"b1",'c',"d1"),('a',"b1",'c',"d2"),('a',"b2",'c',"d1"),('a',"b2",'c',"d2")]

    > 1 *| [(+2)]
    [3]
    > 1 *| [(+)] |* 2
    [3]
    > 1 *|[(+),(-),(*),(^)]|* 2
    [3,-1,2,1]

    > 1 *|Just (,)|* 2
    Just (1,2)

#### Level-2

bra-ket notation:

    > (+1) |$>> [[2]]
    [[3]]
    > [[2]] <<$| (+1)
    [[3]]

    > [Just 1] <<$|(+)|*>> [Just 2]
    [Just 3]
    > [Just 1] <<$|(,)|*>> [Just 2]
    [Just (1,2)]

    > [[1]] <<$|(+)|*>> [[2]] <<$|(-)|*>> [[3]]
    [[0]]

    > foldr (\n acc -> n <<$|(+)|*>> acc) ((.**) 0) [Right (Just 1), Right (Just 2), Right (Just 3)] :: Either () (Maybe Int)
    Right (Just 6)
    > foldr (\n acc -> n <<$|(+)|*>> acc) ((.**) 0) [Right (Just 1), Right Nothing, Right (Just 3)] :: Either () (Maybe Int)
    Right Nothing
    > foldr (\n acc -> n <<$|(+)|*>> acc) ((.**) 0) [Right (Just 1), Right Nothing, Left ()]
    Left ()

cover notation:

    > :t (.**)
    (.**) :: (Applicative f1, Applicative f2) => a -> f1 (f2 a)
    > :t (-*)
    (-*) :: (Applicative f1, Applicative f2) => f1 a -> f1 (f2 a)

    > (.**) 1 :: Maybe [Int]
    Just [1]
    > (-*) (Just 1) :: Maybe [Int]
    Just [1]
    > (.*) [1] :: Maybe [Int]
    Just [1]

cover-braket notation:

    > :t (|**)
    (|**) :: (Applicative f1, Applicative f2) => f1 (f2 (a -> b)) -> a -> f1 (f2 b)

    > [Just 1] <<$|(+)|** 2
    [Just 3]
    > 1 **|(+)|$>> [Just 2]
    [Just 3]
    > 1 **|[Just (+)]|**  2
    [Just 3]
    > 1 **|[Just (+), Just (-), Just (*), Nothing]|** 2
    [Just 3,Just (-1),Just 2,Nothing]

    > :t (|-*)
    (|-*) :: (Applicative f1, Applicative f2) => f1 (f2 (a -> b)) -> f1 a -> f1 (f2 b)
    > :t (|*-)
    (|*-) :: (Applicative f1, Applicative f2) => f1 (f2 (a -> b)) -> f2 a -> f1 (f2 b)

    > [Just 1] <<$|(+)|-* [2]
    [Just 3]
    > [Just 1] <<$|(+)|*- Just 2
    [Just 3]
    >      [1]  -*|(+)|$>> [Just 2]
    [Just 3]
    >   Just 1  *-|(+)|$>> [Just 2]
    [Just 3]
    >   Just 1  *-|[Just (+)]|** 2
    [Just 3]
    >   Just 1  *-|[Just (+)]|*- Just 2
    [Just 3]
    >      [1]  -*|[Just (+)]|*- Just 2
    [Just 3]
    >      [1]  -*|[Just (+), Just (-), Just (*), Nothing]|*- Just 2
    [Just 3,Just (-1),Just 2,Nothing]
    >    [1,2]  -*|[Just (+), Just (-), Just (*), Nothing]|*- Just 2
    [Just 3,Just (-1),Just 2,Nothing,Just 4,Just 0,Just 4,Nothing]

#### Level-3, Level-4 and Level-5

Work well likewise.

### [Traversable](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Traversable.html)

    Prelude> :m DeepControl.Traversable

List, Maybe, Either, Except and Writer monads are all sinkable infinitely.

    > :t sink
    sink :: (Applicative f, Traversable c) => c (f a) -> f (c a)

    > sink $ Just [1]
    [Just 1]
    > sink2 $ Just (Right [1])
    Right [Just 1]

    > sink $ Right [Just 1]
    [Right (Just 1)]
    > sink2 $ Right [Just 1]
    [Just (Right 1)]

So these monads can be deepened.

### [Monad](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Monad.html)

This module enables you to program in Monad for more deeper level than the usual Monad module expresses.
You would soon realize exactly what more deeper level means by reading the example codes below in order.

#### Level-0

```haskell
import DeepControl.Monad ((>-))

plus :: Int -> Int -> Int
plus x y = 
    x >- \a ->   -- (>-) is the level-0 bind function, analogous to (>>=)
    y >- \b ->
    a + b

-- > plus 3 4
-- 7
```

#### Level-2

```haskell
import DeepControl.Applicative ((.**))
import DeepControl.Monad ((>>==))

listlist :: [[String]]             -- List-List monad
listlist = [["a","b"]] >>== \x ->  -- (>>==) is the level-2 bind function, analogous to (>>=)
           [[0],[1,2]] >>== \y ->
           (.**) $ x ++ show y

-- > listlist
-- [["a0","b0"],["a0","b1","b2"],["a1","a2","b0"],["a1","a2","b1","b2"]]
```

```haskell
import DeepControl.Applicative ((|$>), (.*), (.**))
import DeepControl.Monad ((>>), (>>==), (->~))
import Control.Monad.Writer

factorial :: Int ->
             Maybe (Writer [Int] Int)               -- Maybe-Writer monad
factorial n | n < 0  = Nothing
            | n == 0 = (.*) $ tell [0] >> (.*) 1
            | n > 0  = factorial (n-1) >>== \v ->   
                       tell [v] ->~                 -- (->~) is a level-2 cover-sequence function, analogous to (>>)
                       (.**) (n * v)

-- > runWriter |$> factorial 5
-- Just (120,[0,1,1,2,6,24])
-- > factorial (-1)
-- Nothing
```

#### Level-3

```haskell
import DeepControl.Applicative ((|$>>), (.*), (.**), (.***))
import DeepControl.Monad ((>>), (>>>==), (>--~), (-->~))
import Control.Monad.Writer

factorial :: Int ->
             IO (Maybe (Writer [Int] Int))            -- IO-Maybe-Writer monad
factorial n | n < 0  = (.*) Nothing                  
            | n == 0 = (.**) $ tell [0] >> (.*) 1
            | n > 0  = factorial (n-1) >>>== \v ->    -- (>>>==) is the level-3 bind function, analogous to (>>=)
                       print v >--~                   -- (>--~) is a level-3 cover-sequence function, analogous to (>>)
                       tell [v] -->~                  -- (-->~) is a level-3 cover-sequence function too, analogous to (>>)
                       (.***) (n * v)

-- > runWriter |$>> factorial 5
-- 1
-- 1
-- 2
-- 6
-- 24
-- Just (120,[0,1,1,2,6,24])
-- > factorial (-1)
-- Nothing
```
#### Level-4 and Level-5

Work well likewise.

### [Monad-Transformer](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Monad-Trans.html)

#### Level-2

Here is a monad transformer example how to implement Ackermann function improved to stop within a certain limit of time, with ReaderT-IdentityT2-IO-Maybe monad, a level-2 monad-transformation.

```haskell
import DeepControl.Applicative
import DeepControl.Traversable (sink)
import DeepControl.Monad ((>-))
import DeepControl.Monad.Morph ((|*|), (|>|))
import DeepControl.Monad.Trans (transroll2, untransroll2)
import DeepControl.Monad.Trans.Identity (Identity(..), IdentityT(..), IdentityT2(..))
import Control.Monad.Reader
import Control.Monad.Trans.Maybe

import System.Timeout (timeout)

type TimeLimit = Int

ackermannTimeLimit :: TimeLimit -> Int -> Int -> 
                      IO (Maybe Int)                       -- IO-Maybe Monad
ackermannTimeLimit timelimit x y = timeout timelimit (ackermannIO x y)
  where
    ackermannIO :: Int -> Int -> IO Int
    ackermannIO 0 n = (.*) $ n + 1
    ackermannIO m n | m > 0 && n == 0 = ackermannIO (m-1) 1
                    | m > 0 && n > 0  = ackermannIO m (n-1) >>= ackermannIO (m-1)
 
ackermann :: Int -> Int -> 
             ReaderT TimeLimit (IdentityT2 IO Maybe) Int   -- ReaderT-IdentityT2-IO-Maybe monad
ackermann x y = do
    timelimit <- ask
    (|*|) . IdentityT2 $ ackermannTimeLimit timelimit x y  -- lift IO-Maybe function to ReaderT-IdentityT2-IO-Maybe function

calc_ackermann :: TimeLimit -> Int -> Int -> IO (Maybe Int)
calc_ackermann timelimit x y = ackermann x y >- \r -> runReaderT r timelimit
                                             >- runIdentityT2

-- λ> sink $ calc_ackermann 1000 |$> [0..4] |* 4
-- [Just 5,Just 6,Just 11,Just 125,Nothing]

ackermann' :: Int -> Int -> 
              ReaderT TimeLimit (MaybeT IO) Int                 -- ReaderT-MaybeT-IO monad
ackermann' x y = (transroll2 . runIdentityT2) |>| ackermann x y -- You can get usual ReaderT-MaybeT-IO function from ReaderT-IdentityT2-IO-Maybe function

ackermann'' :: Int -> Int -> 
               ReaderT TimeLimit (IdentityT2 IO Maybe) Int       -- ReaderT-IdentityT2-IO-Maybe monad
ackermann'' x y = (IdentityT2 . untransroll2) |>| ackermann' x y -- You can get ReaderT-IdentityT2-IO-Maybe function from usual ReaderT-MaybeT-IO function
```
#### Level-3, Level-4 and Level-5

Work well likewise.

transroll and untransroll:

    Prelude> :m DeepControl.Monad.Trans 
    > :m + Control.Monad.Trans.List Control.Monad.Trans.Maybe
    > :m + DeepControl.Monad.Trans.Identity DeepControl.Monad.Trans.Except DeepControl.Monad.Trans.Writer

    > transroll3 $ ExceptT (Identity (Right [Just 1]))    -- note: type Except e = ExceptT e Identity
    MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))

    > :t ExceptT (Identity (Right [Just 1]))
    ExceptT (Identity (Right [Just 1]))
      :: Num a => ExceptT e Identity [Maybe a]
    > :t transroll3 $ ExceptT (Identity (Right [Just 1]))
    transroll3 $ ExceptT (Identity (Right [Just 1]))
      :: Num a => MaybeT (ListT (ExceptT e Identity)) a

    > untransroll3 $ MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))
    ExceptT (Identity (Right [Just 1]))

    > :t MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))
    MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))
      :: Num a => MaybeT (ListT (ExceptT e Identity)) a
    > :t untransroll3 $ MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))
    untransroll3 $ MaybeT (ListT (ExceptT (Identity (Right [Just 1]))))
      :: Num a => ExceptT e Identity [Maybe a]

### [Monad-Morph](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Monad-Morph.html)

Here is a monad morph example how to use trans-map functions.

```haskell
import DeepControl.Monad.Morph
import Control.Monad.Writer
import Control.Monad.State

-- i.e. :: StateT Int Identity ()
tick    :: State Int ()
tick = modify (+1)

tock                        ::                   StateT Int IO ()
tock = do
    generalize |>| tick     :: (Monad      m) => StateT Int m  ()  -- (|>|) is the level-1 trans-map function, analogous to (|$>)
    lift $ putStrLn "Tock!" :: (MonadTrans t) => t          IO ()

-- λ> runStateT tock 0
-- Tock!
-- ((),1)

-- i.e. :: StateT Int (WriterT [Int] Identity) ()
save    :: StateT Int (Writer  [Int]) ()
save = do
    n <- get
    lift $ tell [n]

program ::                   StateT Int (WriterT [Int] IO) ()
program = replicateM_ 4 $ do
    lift |>| tock
        :: (MonadTrans t) => StateT Int (t             IO) ()
    generalize |>>| save                                        -- (|>>|) is the level-2 trans-map function, analogous to (|$>>)
        :: (Monad      m) => StateT Int (WriterT [Int] m ) ()

-- λ> execWriterT (runStateT program 0)
-- Tock!
-- Tock!
-- Tock!
-- Tock!
-- [1,2,3,4]
```

Here is a monad morph example how to use trans-cover and trans-bind functions.

```haskell
import DeepControl.Monad.Morph ((|>=), (|>>=), (|*|), (|-*|))
import DeepControl.Monad.Trans.Except

import Control.Monad.Trans.Maybe
import Control.Exception (IOException, try)

-----------------------------------------------
-- Level-1 

catchIOError :: IO a -> 
                ExceptT IOException IO a   -- ExceptT-IO monad
catchIOError io = ExceptT $ (try io)

viewFile :: IO ()                          -- IO monad
viewFile = do
    str <- readFile "test.txt"
    putStr str

program :: ExceptT IOException IO ()       -- ExceptT-IO monad
program = (|*|) viewFile |>= catchIOError  -- (|*|) is the level-1 trans-cover function, alias to 'lift' and analogous to (.*)
                                           -- (|>=) is the level-1 trans-bind function, analogous to (>>=)

calc_program :: IO (Either IOException ())
calc_program = runExceptT $ program

-- > calc_program
-- Left test.txt: openFile: does not exist (No such file or directory)

-----------------------------------------------
-- Level-2

viewFile2 :: String -> 
             MaybeT IO ()                        -- MaybeT-IO monad
viewFile2 filename = do
    guard (filename /= "")
    str <- (|*|) $ readFile "test.txt"
    (|*|) $ putStr str

program2 :: String -> 
            (ExceptT IOException (MaybeT IO)) () -- ExceptT-MaybeT-IO monad
program2 filename = 
    (|*|) (viewFile2 filename) |>>= \x ->        -- (|>>=) is the level-2 trans-bind function, analogous to (>>=)
    (|-*|) $ catchIOError x                      -- (|-*|) is a level-2 trans-cover function, analogous to (-*)

calc_program2 :: String -> IO (Maybe (Either IOException ())) 
calc_program2 filename = runMaybeT . runExceptT $ program2 filename

-- > calc_program "test.txt"
-- Just (Left test.txt: openFile: does not exist (No such file or directory))
-- > calc_program ""
-- Nothing
```

#### Level-3, Level-4 and Level-5

Work well likewise.

### [Arrow](https://hackage.haskell.org/package/deepcontrol-0.5.1.0/docs/DeepControl-Arrow.html)