# Prelude extensions

## Sorted lists

The following are versions of standard prelude functions, but intended for sorted lists. The advantage is that they frequently reduce execution time by an O(n). The disadvantage is that the elements have to be members of Ord, and the lists have to be already sorted.

``` -- Eliminates duplicate entries from the list, where duplication is defined
-- by the 'eq' function.  The last value is kept.
sortedNubBy :: (a -> a -> Bool) -> [a] -> [a]
sortedNubBy eq (x1 : xs@(x2 : _)) =
if eq x1 x2 then sortedNubBy eq xs else x1 : sortedNubBy eq xs
sortedNubBy _ xs = xs

sortedNub :: (Eq a) => [a] -> [a]
sortedNub = sortedNubBy (==)

-- Merge two sorted lists into a new sorted list.  Where elements are equal
-- the element from the first list is taken first.
mergeBy :: (a -> a -> Ordering) -> [a] -> [a] -> [a]
mergeBy cmp xs@(x1:xs1) ys@(y1:ys1) =
if cmp x1 y1 == GT
then y1 : mergeBy cmp xs ys1
else x1 : mergeBy cmp xs1 ys
mergeBy _ [] ys = ys
mergeBy _ xs [] = xs

merge :: (Ord a) => [a] -> [a] -> [a]
merge = mergeBy compare
```

## Tuples

It is often necessary to apply functions to either the first or the second part of a pair. This is often considered a form of mapping (like map from Data.List).

``` -- | Apply a function to the first element of a pair
mapFst :: (a -> c) -> (a, b) -> (c, b)
mapFst f (a, b) = (f a, b)

-- | Apply a function to the second element of a pair
mapSnd :: (b -> c) -> (a, b) -> (c, b)
mapSnd f (a, b) = (a, f b)

-- | Apply a function to both elements of a pair
mapPair :: (a -> c, b -> d) -> (a, b) -> (c, d)
mapPair (f, g) (a, b) = (f a, g b)
```

Data.Graph.Inductive.Query.Monad module (section Additional Graph Utilities) contains `mapFst`, `mapSnd`, and also a function `><` corresponding to `mapPair`. Another implementation of these functions in the standard libraries: using `first`, `second`, `***` arrow operations overloaded for functions (as special arrows), see Control.Arrow module, or Arrow HaskellWiki page.

### Treating pairs and lists in the same way

We can define a Pair class which allows us to process both pairs and non-empty lists using the same operator:

```import Control.Arrow ((***))

infixl 4 <**>

class Pair p x y | p -> x, p -> y where
toPair :: p -> (x, y)
(<**>) :: (x -> a -> b) -> (y -> a) -> p -> b
(<**>) f g = uncurry id . (f *** g) . toPair

instance Pair (a, b) a b where
toPair = id

instance Pair [a] a [a] where
toPair l = (head l, tail l)
```

## Matrices

A simple representation of matrices is as lists of lists of numbers:

``` newtype Matrix a = Matrix [[a]] deriving (Eq, Show)
```

These matrices may be made an instance of `Num` (though the definitions of `abs` and `signum` are just fillers):

``` instance Num a => Num (Matrix a) where
Matrix as + Matrix bs = Matrix (zipWith (zipWith (+)) as bs)
Matrix as - Matrix bs = Matrix (zipWith (zipWith (-)) as bs)
Matrix as * Matrix bs =
Matrix [[sum \$ zipWith (*) a b | b <- transpose bs] | a <- as]
negate (Matrix as) = Matrix (map (map negate) as)
fromInteger x = Matrix (iterate (0:) (fromInteger x : repeat 0))
abs m = m
signum _ = 1
```

The `fromInteger` method builds an infinite matrix, but addition and subtraction work even with infinite matrices, and multiplication works as long as either the first matrix is of finite width or the second is of finite height. Applying the linear transformation defined by a matrix to a vector is

``` apply :: Num a => Matrix a -> [a] -> [a]
apply (Matrix as) b = [sum (zipWith (*) a b) | a <- as]
```

## Data.Either extensions

```import Data.Either

either', trigger, trigger_, switch ::  (a -> b) -> (a -> b) -> Either a a -> Either b b

either' f g (Left x) = Left (f x)
either' f g (Right x) = Right (g x)

trigger f g (Left x) = Left (f x)
trigger f g (Right x) = Left (g x)

trigger_ f g (Left x) = Right (f x)
trigger_ f g (Right x) = Right (g x)

switch f g (Left x) = Right (f x)
switch f g (Right x) = Left (g x)

sure :: (a->b) -> Either a a -> b
sure f = either f f

sure' :: (a->b) -> Either a a -> Either b b
sure' f = either' f f
```