ranged-list

What’s this

This package provides lists whose lengths are determined by the type and lists whose ranges of lengths are determined by the type.

sample1 :: LengthL 3 Integer
sample1 = 1 :. 2 :. 3 :. NilL

sample2 :: LengthR 3 Integer
sample2 = NilR :+ 1 :+ 2 :+ 3

sample3 :: RangeL 2 5 Integer
sample3 = 1 :. 2 :. 3 :.. 4 :.. NilL

sample4 :: RangeR 2 5 Integer
sample4 = NilR :++ 1 :++ 2 :+ 3 :+ 4

LengthL 3 Integer and LengthR 3 Integer are lists who have just 3 Integer. RangeL 2 5 Integer and RangeR 2 5 Integer are lists whose element numbers are 2 at minimum and 5 at maximum. LengthL 3 Integer and RangeL 2 5 Integer are pushed or poped a element from left. LengthR 3 Integer and RangeR 2 5 Integer are pushed or poped a element from right.

Motivation

Suppose you want to take elements from list. You can use take like following.

xs = take 3 "Hello, world!"

The length of xs is lesser or equal 3. But you cannot use this knowledge when you write next code. You should check the argument of a next function.

fun :: [Char] -> ...
fun [] = ...
fun [x] = ...
fun [x, y] = ...
fun [x, y, z] = ...
fun _ = error "bad argument"

If you use LengthL 3 Char, you don’t need to mind the argument has more than 3 elements.

fun :: LengthL 3 Char -> ...
fun (x :. y :. z :. NilL) = ...

LengthL

To make rectangles from a number list

Suppose you want to make a value which represent a rectangle. You have a number list. The numbers are a left border, a top border, a width and a height of a rectangle in order. The numbers of the first rectangle are followed by the numbers of a second rectangle, and the numbers of the second rectangle are followed by the numbers of a third rectangle, and so on.

[left1, top1, width1, height1, left2, top2, width2, height2, left3, ...]

The list of numbers defined above are covert to a following list.

[Rect left1 top1 width1 height1, Rect left2 top2 width2 height2, Rect left3 ...]

The code is following. (View sample/rectangle.hs)

import Data.Length.Length

data Rect = Rect {
	left :: Double, top :: Double,
	width :: Double, height :: Double } derivins Show

makeRect :: Length 4 Double -> Rect
makeRect (l :. t :. w :. h :. NilL) = Rect l t w h

main :: IO ()
main = print $ map makeRect . fst $ chunksL [3, 5, 15, 2, 8, 4, 1, 9, 3, 5]

The function chunksL return a value of type ([LengthL n a], RangeL 0 (n - 1) a). The first value of this tuple is a list of n elements of type a. And the second value of this tuple is rest elements. The number of the rest elements is 0 at minimum and n - 1 at maximum.

Try running.

% stack ghc sample/rectangle.hs
% ./sample/rectangle
[Rect {left = 3.0, top = 5.0, width = 15.0, height = 2.0},
Rect {left = 8.0, top = 4.0, width = 1.0, height = 9.0)}

To take Word64 from bit list

Let’s define function to take a 64 bit word from bit list. (View sample/word64.hs) The language extensions and the import list are following.

{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE MonoLocalBinds #-}
{-# LANGUAGE DAtaKinds, TypeOperators #-}
{-# LANGUAGE FlexibleContexts #-}
{-# OPTIONS_GHC -Wall -fno-warn-tabs #-}

import GHC.TypeNats
import Data.Foldable
import Data.List.Length
import Data.List.Range
import Data.Bits
import Data.Word
import Numeric

You define function takeL to take n elements from list.

takeL :: (LoosenLMax 0 (n - 1) n, Unfoldr 0 n n, ListToLengthL n) =>
	a -> [a] -> LengthL n a
takeL d = either ((`fillL` d) . loosenLMax) fst . splitL

The function splitL split a list and get n element lengthed list (LengthL n a) and a rest of the list. If the list does not contain enough elements, then it returns a left value. It is a list of type RangeL 0 (n - 1) a. The function loosenLMax convert the type RangeL 0 (n - 1) into RangeL 0 n. And the function fillL fill the list with default value d to get a list LengthL n a. Try it.

% stack ghci sample/word64.hs
> :set -XDataKinds
> takeL '@' "Hello, world!" :: LengthL 5 Char
'H' :. ('e' :. ('l' :. ('l' :. ('o' :. NilL))))
> takeL 'W' "Hi!" :: LengthL 5 Char
'H' :. ('i' :. ('!' :. ('@' :. ('@' :. NilL))))

You define data type which represent a bit as follow.

data Bit = O | I deriving Show

boolToBit :: Bool -> Bit
boolToBit = \case False -> O; True -> I

bitToNum63 :: (Num n, Bits n) => Bit -> n
bitToNum63 = \case O -> 0; I -> 1 `shiftL` 63

O is 0 and I is 1. Function boolToBit converts a value of Bool into a value of Bit. Function bitToNum63 converts a value of Bit into a number. It converte the bit as a 63rd bit.

You define the function which convert a bit list into 64 bit word.

bitsToWord64 :: LengthL 64 Bit -> Word64
bitsToWord64 = foldl' (\w b -> w `shiftR` 1 .|. bitToNum63 b) 0

It gets a bit from the left end. It put the bit on a 63rd position of a 64 bit word. Then it gets a next bit. It shifts 64 bit word to the right. And it put the bit on a 63rd position of a 64 bit word. It continue in the same way.

You define the function which take 64 bit word from a bit list expressed as string.

takeWord64 :: String -> Word64
takeWord64 = bitsToWord64 . takeL O . (boolToBit . (== '*') <$>)

The argument of this function is a string. The string represent a bit sequence. Character '*' is 1 and character '.' is 0.

You define sample string and try it in function main.

sample1, sample2 :: String
sample1 = "...*..*..*...........*...**********...*************............******"
sample2 = "...*..*..*...........*.."

main :: IO ()
main = do
	putStrLn $ takeWord64 sample1 `showHex` ""
	putStrLn $ takeWord64 sample2 `showHex` ""

Try it.

% stack ghc sample/word64.hs
% ./sample/word64
8007ffc7fe200248
200248

LengthR

To push and pop from right

A value of the type LengthR n a is a list of values of the type a. The length of the list is n. And you can push and pop an element from right. Try it. (view sample/LengthR.hs)

{-# LANGUAGE DataKinds #-}
{-# OPTIONS_GHC -Wall -fno-warn-tabs #-}

module LengthR where

import Data.List.Length

hello :: LengthR 5 Char
hello = NilR :+ 'h' :+ 'e' :+ 'l' :+ 'l' :+ 'o'

The value hello is a list of characters which length is 5. Let's push the character '!' from right.

% stack ghci sample/LengthR.hs
> hello
((((NilR :+ 'h') :+ 'e') :+ 'l') :+ 'l') :+ 'o'
> hello :+ '!'
(((((NilR :+ 'h') :+ 'e') :+ 'l') :+ 'l') :+ 'o') :+ '!'

To show 4 points of rectangles

function fourPoints and headers

You want to calculate four points of rectangle from the left-top point, width and height of the rectangle. You define function fourPoints. (View sample/fourPointsOfRect.hs)

fourPoints :: LengthR 4 Double -> LengthR 4 (Double, Double)
fourPoints (NilR :+ l :+ t :+ w :+ h) =
	NilR :+ (l, t) :+ (l + w, t) :+ (l, t + h) :+ (l + w, t + h)

You add language extensions and modules to import.

{-# LANGUAGE BlockArguments, LambdaCase #-}
{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
{-# LANGUAGE DataKinds, TypeOperators #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE MultiParamTypeClasses, FlexibleContexts, FlexibleInstances,
	UndecidableInstances #-}
{-# OPTIONS_GHC -Wall -fno-warn-tabs -fplugin=Plugin.TypeCheck.Nat.Simple #-}

import GHC.TypeNats
import Control.Monad.Fix
import Control.Monad.Catch
import Data.List.Length
import Text.Read

Try it.

% stack ghci sample/fourPointsOfRect.hs
> fourPoints $ NilR :+ 300 :+ 200 :+ 50 :+ 30
(((NilR :+ (300.0,200.0)) :+ (350.0,200.0)) :+ (300.0,230.0)) :+ (350.0,230.0)

to input values interactively

You want to input values of a left bound, a top bound, a width and a height interactively. You want to delete the last value and reinput a new value. First of all, you define two data type, DeleteOr a and NothingToDeleteException.

data DeleteOr a = Delete | Value a deriving Show
data NothingToDeleteException = NothingToDeleteException deriving Show
instance Exception NothingToDeleteException

And you define the function getElems as a class function.

class GetElems n v where
	getElems :: MonadThrow m =>
		LengthR n a -> m (Maybe (DeleteOr a)) -> m (LengthR (n + v) a)

instance GetElems 0 0 where getElems NilR _ = pure NilR

instance {-# OVERLAPPABLE #-} 1 <= n => GetElems n 0 where
	getElems xs@(_ :+ _) _ = pure xs

instance {-# OVERLAPPABLE #-} GetElems 1 (v - 1) => GetElems 0 v where
	getElems NilR gt = gt >>= \case
		Nothing -> getElems NilR gt
		Just Delete -> throwM NothingToDeleteException
		Just (Value x) -> getElem @1 @(v - 1) (NilR :+ x) gt

instance {-# OVERLAPPABLE #-}
	(1 <= n, GetElems (n - 1) (v + 1), GetElems (n + 1) (v - 1)) =>
	GetElems n v where
	getElems xa@(xs :+ _) gt = gt >>= \case
		Nothing -> getElems xa gt
		Just Delete -> getElems @(n - 1) @(v + 1) xs gt
		Just (Value x) -> getElems @(n + 1) @(v - 1) (xa :+ x) gt

class GetElems n v

The class function getElems has two arguments. The first argument is a list of values which are already inputed. The second argument is a monad which returns 3 kinds of values, a value which represents to delete, a new value to push to the list or a value which represents to do nothing.

instance GetElems 0 0

n == 0 and v == 0 means that the function getElems get a list of no elements and return a list of no elements.

instance GetElems n 0

v == 0 means that the function getElems get a list and return the list as it is.

instance GetElems 0 v

n == 0 means that there are no already inputed elements. The monad returns 3 kind of values. If it returns Nothing, then it rerun the whole as getElems NilR gt. If it returns Just Delete, then NothingToDeleteException occurs. If it returns Just (Value x), then it set the already-inputed elements to NilR :+ x and rerun the whole.

instance GetElems n v

The monad gt returns 3 kind of values. If it returns Nothing, then rerun the whole as getElems xa gt. If it returns Just Delete, then it remove an element from the already-inputed list and rerun the whole. If it returns Just (Value x), then it set the already-inputed elements to xa :+ x and rerun the whole.

to try it

Try it.

% stack ghci sample/fourPointsOfRect.hs
> :set -XDataKinds -XBlockArguments -XLambdaCase
> getElems NilR (Just . Value <$> getLine) :: IO (LengthR 3 String)
foo
bar
baz
((NilR :+ "foo") :+ "bar") :+ "baz"
> gt = (<$> getLine) \case "" -> Nothing; "d' -> Just Delete; s -> Just (Value s)
> getElems NilR gt :: IO (LengthR 3 String)
foo
bar
d
boo

baz
((NilR :+ "foo") :+ "boo") :+ "baz"
> getElems NilR gt :: IO (LengthR 3 String)
foo
bar
d
d
hoge
piyo
baz
((NilR :+ "hoge") :+ "piyo") :+ "baz"
> getElems NilR gt :: IO (LengthR 3 String)
foo
bar
d
d
d
*** Exception: NothingToDeleteException

function titles

You define the function titles which show values as string with title.

titles :: (Show a, Applicative (LengthR n)) =>
	Int -> LengthR n String -> LengthR n a -> LengthR n String
titles n ts xs = (\t x -> t ++ replicate (n - length t) ' ' ++ ": " ++ show x)
	<$> ts <*> xs

Try it.

% stack ghci sample/fourPointsOfRect.hs
> titles 5 (NilR :+ "foo" :+ "bar" :+ "baz") (NilR :+ 123 :+ 456 :+ 789)
((NilR :+ "foo  : 123") :+ "bar  : 456") :+ "baz  : 789"

function printResult

You define the function printResult which show values expressing a rectangle and 4 points of rectangle.

printResult :: LengthR 4 Double -> IO ()
printResult r = do
	putStrLn ""
	putStrLn `mapM_` titles 6 t r; putStrLn ""
	putStrLn `mapM_` titles 12 u (fourPoints r); putStrLn ""
	where
	t = NilR :+ "left :+ "top" :+ "width" :+ "height"
	u = NIlR :+ "left-top" :+ "right-top" :+ "left-bottom" :+ "right-bottom"

Try it.

% stack ghci sample/fourPointsOfRect.hs
> printResult $ NilR :+ 300 :+ 200 :+ 70 :+ 50

left  : 300.0
top   : 200.0
width : 70.0
height: 50.0

left-top    : (300.0,200.0)
right-top   : (370.0,200.0)
left-bottom : (300.0,250.0)
right-bottom: (370.0,250.0)

function getRect

You define the function getRect which gets user input to make rectangle.

getRect :: forall n . GetElems n (4 - n) =>
	LengthR n Double -> IO (LengthR 4 Double)
getRect xs = (<$) <$> id <*> printRect =<<
	getElems @n @(4 - n) xs ((<$> getLine) \case
		"d" -> Just Delete; l -> Value <*> readMaybe l)
	`catch`
	\(_ :: NothingToDeleteException) ->
		putStrLn *** Nothing to delete." >> getRect @0 NilR

It gets a user input with getLine. If it is "d", then it deletes the last input. If there are nothing to delete, then NothingToDeleteException occur. It catches this exception and shows error message and rerun getRect.

function main

You define function main.

main :: IO ()
main = getRect NilR >>= fix \go xa@(xs :+ _) -> getLine >>= \case
	"q" -> pure ()
	"d" -> go =<< getRect xs
	_ -> putStrLn "q or d" >> go xa

It call function getRect with list of 0 elements (NilR). And it repeats function getRect with list of 4 - 1 elements (xs) if you input "d".

% stack ghc sample/fourPointsOfRect.hs
% ./sample/fourPointsOfRect
500
300
75
50

left  : 500.0
top   : 300.0
width : 75.0
height: 50.0

left-top    : (500.0,300.0)
right-top   : (575.0,300.0)
left-bottom : (500.0,350.0)
right-bottom: (575.0,350.0)

d
d
125
100

left  : 500.0
top   : 300.0
width : 125.0
height: 100.0

left-top    : (500.0,300.0)
right-top   : (625.0,300.0)
left-bottom : (500.0,400.0)
right-bottom: (625.0,400.0)

d
d
d
d
d
*** Nothing to delete.
2000
1500
90
50

left  : 2000.0
top   : 1500.0
width : 90.0
height: 50.0

left-top    : (2000.0,1500.0)
right-top   : (2090.0,1500.0)
left-bottom : (2000.0,1550.0)
right-bottom: (2090.0,1550.0)

q

RangeL and RangeR

To specify the range of a number of elements of a list

You can specify the range of a number of elements of a list. There is a data type RangeL n m a. It represents a list which have a type a element. And its length is n at minimum and m at maximum.

% stack ghci
> :module Data.List.Range
> :set -XDataKinds
> 'h' :. 'e' :. 'l' :. 'l' :.. 'o' :.. NilL :: RangeL 3 8 Char
'h' :. ('e' :. ('l' :. ('l' :.. ('o' :.. NilL))))

To get passwords

Suppose you want to get a password whose length is 8 at minimum and 127 at maximum. First of all, you define headers.

{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE FlexibleContexts #-}
{-# OPTIONS_GHC -Wall -fno-warn-tabs #-}

import Data.List.Range
import System.IO

import qualified Data.ByteString.Char8 as BSC

You define type Password.

type Password = RangeL 8 127 Char

It is a list of Char. Its length is 8 at minimum and 127 at maximum.

You define a function getRangedString. It recieves a user input. It return a just value if the length of the input is within range. It return a nothing value if the length of the input is out of range.

getRangedPassword :: Unfoldr 0 n m => IO (Maybe (RangeL n m Char))
getRangedPassword = do
	e <- hGetEcho stdin
	hSetEcho stdin False
	unfoldrMRangeMaybe ((/= '\n') <$> hLookAhead stdin) getChar
		<* hSetEcho stdin e

It makes echo of stdin off. It gets characters until you input '\n'. And it makes echo of stdin on.

% stack ghci sample/password.hs
> :set -XDataKinds
> getRangedPassword :: IO (Maybe Password)
(Input "foobarbaz")
Just ('f' :. ('o' :. ('o' :. ('b' :. ('a' :. ('r' :. ('b' :. ('a' :. ('z' :..NilL)))))))))
> getRangedPassword :: IO (Maybe Password)
(Input "foo")
Nothing
> getRangedPassword :: IO (Maybe (RangeL 2 5 Char))
(Input "foobar")
Nothing
> r

You want to convert a value of type Password into a value of ByteString. You can use other packages if you get password as a value of ByteString.

passwordToByteString :: Password -> BSC.ByteString
passwordToByteString = foldr BSC.cons ""

You define function main to try it.

main :: IO ()
main = do
	p <- getRangedPassword
	print p
	maybe (eror "bad password length") BSC.putStrLn $ passwordToByteString <$> p

Try it.

% stack ghc sample/password.hs
% ./sample/password
(Input "foobarbaz")
Just ('f' :. ('o' :. ('o' :. ('b' :. ('a' :. ('r' :. ('b' :. ('a' :. ('z' :.. NilL)))))))))
foobarbaz

Finger Tree

The next example is Finger Tree.

Finger Trees: A Simple General-purpose Data Structure

Language Extension and Import List

Let’s make headers.

{-# LANGUAGE ScopedTypeVariables, TypeApplications, InstanceSigs #-}
{-# LANGUAGE DataKinds, TypeOperators #-}You
{-# LANGUAGE MultiParamTypeClasses, FlexibleContexts, FlexibleInstances,
	UndecidableInstances #-}
{-# OPTIONS_GHC -Wall -fno-warn-tabs -fplugin=Plugin.TypeCheck.Nat.Simple #-}

import GHC.TypeNats
import Data.List.Range

Types

You can describe Finger Tree as follows.

data FingerTree a
	= Enpty | Single a
	| Deep (DigitL a) (FingerTree (Node a)) (DigitR a)
	deriving Show

type Node = RangeL 2 3
type DigitL = RangeL 1 4
type DigitR = RangeR 1 4

A list of type Node a contains two or three elements of type a. A list of type DigitL a contains one elements at minimum and four elements at maximum. A list of type DigitR a contains the same number of elements as DigitL a. But you can push and pop a element from right.

To push from left

You define the function which Add a new element to the left of the sequence. First of all you define the function to push an element to a list of type DigitL a.

infixr 5 <||

(<||) :: a -> DigitL a -> Either (DigitL a) (DigitL a, Node a)
a <|| b :. NilL = Left $ a :. b :.. NilL
a <|| b :. c :.. NilL = Left $ a :. b :.. c :.. NilL
a <|| b :. c :.. d :.. NilL = Left $ a :. b :.. c :.. d :.. NilL
a <|| b :. c :.. d :.. e :.. NilL =
	Right (a :. b :.. NilL, c :. d :. e :.. NilL)

If the original list has fewer elements than four, then it return a left value list which contains the added value. If the original list has just four elements, then it returns a right value tuple which contain the value of type DigitL a and the value of type Node a.

You can define the function which add a new element to the left of the sequence.

infixr 5 <|

(<|) :: a -> FingerTree a -> FingerTree a
a <| Empty = Single a
a <| Single a = Deep (a :. NilL) Empty (NilR :+ b)
a <| Deep pr m sf = case a <|| pr of
	Left pr' -> Deep pr' m sf
	Right (pr', n3) -> Deep pr' (n3 <| m) sf

It pushes three of the elements as a Node, leaving two behind.

You also require the liftings of <|.

infixr 5 <|.

(<|.) :: Foldable t => t a -> FingerTree a -> FingerTree a
(<|.) = flip $ foldr (<|)

To make finger tree from a list or other foldable structure, you define a function toTree.

toTree :: Foldable t => t a -> FingerTree a
toTree = (<|. Empty)

To push from right

Adding to the right end of the sequence is the mirror image of the above.

infixl 5 ||>, |>, |>.

(||>) :: DigitR a -> a -> Either (DigitR a) (Node a, DigitR a)
NilR :+ a ||> b = Left $ NilR :++ a :+ b
NilR :++ a :+ b ||> c = Left $ NilR :++ a :++ b :+ c
NIlR :++ a :++ b :+ c ||> d = Left $ NilR :++ a :++ b :++ c :+ d
NilR :++ a :++ b :++ c :+ d ||> e =
	Right (a :. b :. c :.. NilL, NilR :++ d :+ e)

(|>) :: FingerTree a -> a -> FingerTree a
Empty |> a = Single a
Single a |> b = Deep (a :. NilL) Empty (NilR :+ b)
Deep pr m sf |> a = case sf ||> a of
	Left sf' -> Deep pr m sf'
	Right (n3, sf') -> Deep pr (m |> n3) sf'

(|>.) :: Foldable t => FingerTree a -> t a -> FingerTree a
(|>.) = foldl (|>)

To pop from left

To deconstruct a sequence, you define a function uncons.

uncons :: FingerTree a -> Maybe (a, FingerTree a)
uncons Empty = Nothing
uncons (Single x) = Just (x, Empty)
uncons (Deep (a :. pr') m sf) = Just (a, deepL pr' m sf)

deepL :: RangeL 0 3 a -> FingerTree (Node a) -> DigitR a -> FingerTree a
deepL NilL m sf = case uncons m of
	Nothing -> toTree sf
	Just (n, m') -> Deep (loosenL n) m' sf
deepL (a :.. pr) m sf = Deep (loosenL $ a :. pr) m sf

Since the prefix pr of a Deep tree contains at least one element, you can get its head. However, the tail of the prefix may be empty, and thus unsuitable as a first argument to the Deep constructor. Hence you define a smart constructor that differs from Deep by allowing the prefix to contain zero to three elements, and in the empty case uses a uncons of the middle tree to construct a tree of the correct shape.

Concatenation

First of all you define a function which devide a list into a list of Node. The original list has 3 elements at minimum and 12 elements at maximum. The returned list has 1 node at minimum and 4 nodes at maximum. The function has a type like the following.

fun :: RangeL 3 12 a -> RangeL 1 4 (Node a)

You can define a more general function like the following.

fun :: RangeL 3 m a -> RangeL 1 w (Node a)

m is 3 times w.

You define a class.

class Nodes m w where nodes :: RangeL 3 m a -> RangeL 1 w (Node a)

And you define instance when m is 3 and w is 1.

instance Nodes 3 1 where nodes = (:. NilL) . loosenL	

And you define instance of general case.

instance {-# OVERLAPPABLE #-} (2 <= w, Nodes (m - 3) (w - 1)) => Nodes m w where
	nodes :: forall a . RangeL 3 m a -> RangeL 1 w (Node a)
	nodes (a :. b :. c :. NilL) = (a :. b :. c :.. NilL) :. NilL
	nodes (a :. b :. c :. d :.. NilL) =
		(a :. b :. NilL) :. (c :. d :. NilL) :.. NilL
	nodes (a :. b :. c :. d :.. e :.. NilL) =
		(a :. b :. c :.. NilL) :. (d :. e :. NilL) :.. NilL
	nodes (a :. b :. c :. d :.. e :.. f :.. xs) =
		(a :. b :. c :.. NilL) .:..
			nodes @(m - 3) @(w - 1) (d :. e :. f :. xs)

Try it.

% stack ghci sample/fingertree.hs
> :set -XTypeApplications -XDataKinds
> xs = 1 :. 2 :. 3 :. 4 :.. 5 :.. 6 :.. 7 :.. 8 :.. NilL :: RangeL 3 12 Integer
> nodes @12 @4 xs
(1 :. (2 :. (3 :.. NilL))) :. ((4 :. (5 :. (6 :.. NilL))) :.. ((7 :. (8 :. NilL)) :.. NilL))
> :type it
it :: Num a => RangeL 1 4 (Node a)

You can combine the two digit argument into a list of Nodes with the function nodes. You can obtain a recursive function by generalizing the concatenation function to take an additional list of elements.

app3 :: FingerTree a -> RangeL 1 4 a -> FingerTree a -> FingerTree a
app3 Empty m xs = m <|. xs
app3 xs m Empty = xs |>. m
app3 (Single x) m xs = x <| m <|. xs
app3 xs m (Single x) = xs |>. m |> x
app3 (Deep pr1 m1 sf1) m (Deep pr2 m2 sf2) =
	Deep pr1 (app3 m1 (nodes $ sf1 ++.. m ++. pr2) m2) sf2

To concatenate two finger trees, you take a head element from a second sequence.

(><) :: FingerTree a -> FingerTree a -> FingerTree a
l >< r = case uncons r of Nothing -> l; Just (x, r') -> app3 l (x :. NilL) r'