1. PROGRAMMING IN HASKELL
Odds and Ends, and Type and Data definitions
Based on lecture notes by Graham Hutton
The book “Learn You a Haskell for Great Good”
(and a few other sources)

2. Higher order functions
Remember that functions can also be inputs:
applyTwice :: (a -> a) -> a -> a
applyTwice f x = f (f x)
After loading, we can use this with any function:
ghci> applyTwice (+3) 10 16
ghci> applyTwice (++ " HAHA") "HEY"
"HEY HAHA HAHA"
ghci> applyTwice ("HAHA " ++) "HEY"
"HAHA HAHA HEY"
ghci> applyTwice (3:) [1]
[3,3,1]

3. Useful functions: zipwith
zipWith is a default in the prelude, but if we were coding it, it would look like this:
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
zipWith _ [] _ = []
zipWith _ _ [] = []
zipWith f (x:xs) (y:ys) =
f x y : zipWith' f xs ys
Look at declaration for a bit…

4. Useful functions: zipwith
Using zipWith:
ghci> zipWith (+) [4,2,5,6] [2,6,2,3]
[6,8,7,9]
ghci> zipWith max [6,3,2,1] [7,3,1,5]
[7,3,2,5]
ghci> zipWith (++) ["foo ", "bar ", "baz "]
["fighters", "hoppers", "aldrin"]
["foo fighters","bar hoppers","baz aldrin"]
ghci> zipWith' (*) (replicate 5 2) [1..]
[2,4,6,8,10]
ghci> zipWith' (zipWith' (*))
[[1,2,3],[3,5,6],[2,3,4]]
[[3,2,2],[3,4,5],[5,4,3]]
[[3,4,6],[9,20,30],[10,12,12]]

5. Useful functions: map
The function map applies a function across a list:
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs
ghci> map (+3) [1,5,3,1,6]
[4,8,6,4,9]
ghci> map (++ "!") ["BIFF", "BANG", "POW"]
["BIFF!","BANG!","POW!"]
ghci> map (replicate 3) [3..6]
[[3,3,3],[4,4,4],[5,5,5],[6,6,6]]
ghci> map (map (^2)) [[1,2],[3,4,5,6],[7,8]]
[[1,4],[9,16,25,36],[49,64]]

6. Useful functions: filter
The function fliter:
filter :: (a -> Bool) -> [a] -> [a]
filter _ [] = []
filter p (x:xs)
| p x = x : filter p xs
| otherwise = filter p xs
ghci> filter (>3) [1,5,3,2,1,6,4,3,2,1]
[5,6,4]
ghci> filter (==3) [1,2,3,4,5]
[3]
ghci> filter even [1..10]
[2,4,6,8,10]

7. Using filter: quicksort!
quicksort :: (Ord a) => [a] -> [a]
quicksort [] = []
quicksort (x:xs) =
let smallerSorted = quicksort (filter (<=x) xs)
biggerSorted = quicksort (filter (>x) xs)
in smallerSorted ++ [x] ++ biggerSorted
(Also using let clause, which temporarily binds a function in the local context. The function actually evaluates to whatever “in” is.)

8. Exercise Write a function myZip :: [a] -> [b] -> [(a, b)]
which zips two lists together:
myZip [1,2,3] "abc" = [(1, 'a'), (2, 'b'), (3, 'c')]
(If one list is smaller, just go ahead and stop whenever one of them ends.)
Hint: I’d do this with recursion! Just don’t use the included function zip, since that is cheating.

9. Conditional Expressions
As in most programming languages, functions can be defined using conditional expressions.
abs :: Int  Int
abs n = if n  0 then n else -n
abs takes an integer n and returns n if it is non-negative and -n otherwise.

10. Conditional expressions can be nested:
signum :: Int  Int
signum n = if n < 0 then -1 else
if n == 0 then 0 else 1
Note:
In Haskell, conditional expressions must always have an else branch, which avoids any possible ambiguity issues.

11. As previously, but using guarded equations.
As an alternative to conditionals, functions can also be defined using guarded equations.
abs n | n  = n
| otherwise = -n
As previously, but using guarded equations.

12. Guarded equations can be used to make definitions involving multiple conditions easier to read:
signum n | n < = -1
| n == 0 = 0
| otherwise = 1
Note:
The catch all condition otherwise is defined in the prelude by otherwise = True.

13. not maps False to True, and True to False.
Pattern Matching
Many functions have a particularly clear definition using pattern matching on their arguments.
not :: Bool  Bool
not False = True
not True = False
not maps False to True, and True to False.

14. can be defined more compactly by
Functions can often be defined in many different ways using pattern matching. For example
(&&) :: Bool  Bool  Bool
True && True = True
True && False = False
False && True = False
False && False = False
can be defined more compactly by
True && True = True
_ && _ = False

15. However, the following definition is more efficient, because it avoids evaluating the second argument if the first argument is False:
True && b = b
False && _ = False
Note:
The underscore symbol _ is a wildcard pattern that matches any argument value.

16. Patterns are matched in order
Patterns are matched in order. For example, the following definition always returns False:
_ && _ = False
True && True = True
Patterns may not repeat variables. For example, the following definition gives an error:
b && b = b
_ && _ = False

17. List Patterns
Internally, every non-empty list is constructed by repeated use of an operator (:) called “cons” that adds an element to the start of a list.
[1,2,3,4]
Means 1:(2:(3:(4:[]))).

18. Functions on lists can be defined using x:xs patterns.
head :: [a]  a
head (x:_) = x
tail :: [a]  [a]
tail (_:xs) = xs
head and tail map any non-empty list to its first and remaining elements.

19. x:xs patterns only match non-empty lists:
Note:
x:xs patterns only match non-empty lists:
> head []
Error
x:xs patterns must be parenthesised, because application has priority over (:). For example, the following definition gives an error:
head x:_ = x

20. Lambda Expressions
Functions can be constructed without naming the functions by using lambda expressions.
x  x+x
the nameless function that takes a number x and returns the result x+x.

21. Note:
The symbol  is the Greek letter lambda, and is typed at the keyboard as a backslash \.
In mathematics, nameless functions are usually denoted using the  symbol, as in x  x+x.
In Haskell, the use of the  symbol for nameless functions comes from the lambda calculus, the theory of functions on which Haskell is based.

22. is more naturally defined by
Lambda expressions are useful when defining functions that return functions as results.
For example:
const :: a  b  a
const x _ = x
is more naturally defined by
const :: a  (b  a)
const x = _  x

23. Lambda expressions can be used to avoid naming functions that are only referenced once.
For example:
odds n = map f [0..n-1]
where
f x = x*2 + 1
can be simplified to
odds n = map (x  x*2 + 1) [0..n-1]

24. Sections
An operator written between its two arguments can be converted into a curried function written before its two arguments by using parentheses.
For example:
> 1+2 3
> (+) 1 2

25. This convention also allows one of the arguments of the operator to be included in the parentheses.
For example:
> (1+) 2 3
> (+2) 1
In general, if  is an operator then functions of the form (), (x) and (y) are called sections.

26. Why Are Sections Useful?
Useful functions can sometimes be constructed in a simple way using sections. For example:
- successor function
- reciprocation function
- doubling function
- halving function
(1+)
(*2)
(/2)
(1/)

27. Exercise
Consider a function safetail that behaves in the same way as tail, except that safetail maps the empty list to the empty list, whereas tail gives an error in this case. Define safetail using:
(a) a conditional expression;
(b) guarded equations;
(c) pattern matching.
Hint: the library function null :: [a]  Bool can be used to test if a list is empty.

28. Higher order functions
Last time we started discussing higher order functions, or functions that take other functions are input.
These are heavily useful, and are one of the strengths of Haskell!
Let’s see a few more…

29. The Foldr Function
A number of functions on lists can be defined using the following simple pattern of recursion:
f [] = v
f (x:xs) = x  f xs
f maps the empty list to some value v, and any non-empty list to some function  applied to its head and f of its tail.

30. v = 0 v = 1 v = True For example: sum [] = 0 sum (x:xs) = x + sum xs
 = +
product [] = 1
product (x:xs) = x * product xs
v = 1
 = *
and [] = True
and (x:xs) = x && and xs
v = True
 = &&

31. The higher-order library function foldr (fold right) encapsulates this simple pattern of recursion, with the function  and the value v as arguments.
For example:
sum = foldr (+) 0
product = foldr (*) 1
or = foldr (||) False
and = foldr (&&) True

32. Foldr itself can be defined using recursion:
foldr :: (a  b  b)  b  [a]  b
foldr f v [] = v
foldr f v (x:xs) = f x (foldr f v xs)
However, it is best to think of foldr non-recursively, as simultaneously replacing each (:) in a list by a given function, and [] by a given value.

33. For example: = = = = Replace each (:) by (+) and [] by 0. sum [1,2,3]
foldr (+) 0 [1,2,3] =
foldr (+) 0 (1:(2:(3:[]))) =
1+(2+(3+0)) = 6 =
Replace each (:)
by (+) and [] by 0.

34. For example: = = = = Replace each (:) by (*) and [] by 1.
product [1,2,3]
foldr (*) 1 [1,2,3] =
foldr (*) 1 (1:(2:(3:[]))) =
1*(2*(3*1)) = 6 =
Replace each (:)
by (*) and [] by 1.

35. Other Foldr Examples
Even though foldr encapsulates a simple pattern of recursion, it can be used to define many more functions than might first be expected.
Recall the length function:
length :: [a]  Int
length [] = 0
length (_:xs) = 1 + length xs

36. Replace each (:) by _ n  1+n and [] by 0.
For example:
length [1,2,3]
length (1:(2:(3:[]))) =
1+(1+(1+0)) = 3 =
Replace each (:) by _ n  1+n and [] by 0.
Hence, we have:
length = foldr (_ n  1+n) 0

37. Replace each (:) by x xs  xs ++ [x] and [] by [].
Now recall the reverse function:
reverse [] = []
reverse (x:xs) = reverse xs ++ [x]
For example:
Replace each (:) by x xs  xs ++ [x] and [] by [].
reverse [1,2,3]
reverse (1:(2:(3:[]))) =
(([] ++ [3]) ++ [2]) ++ [1] =
[3,2,1] =

38. Replace each (:) by (:) and [] by ys.
Hence, we have:
reverse =
foldr (x xs  xs ++ [x]) []
Finally, we note that the append function (++) has a particularly compact definition using foldr:
Replace each (:) by (:) and [] by ys.
(++ ys) = foldr (:) ys

39. Why Is Foldr Useful?
Some recursive functions on lists, such as sum, are simpler to define using foldr.
Properties of functions defined using foldr can be proved using algebraic properties of foldr, such as fusion and the banana split rule.
Advanced program optimizations can be simpler if foldr is used in place of explicit recursion.

40. Other Library Functions
The library function (.) returns the composition of two functions as a single function.
(.) :: (b  c)  (a  b)  (a  c)
f . g = x  f (g x)
For example:
odd :: Int  Bool
odd = not . even

41. The library function all decides if every element of a list satisfies a given predicate.
all :: (a  Bool)  [a]  Bool
all p xs = and [p x | x  xs]
For example:
> all even [2,4,6,8,10]
True

42. Dually, the library function any decides if at least
one element of a list satisfies a predicate.
any :: (a  Bool)  [a]  Bool
any p xs = or [p x | x  xs]
For example:
> any isSpace "abc def"
True

43. The library function takeWhile selects elements from a list while a predicate holds of all the elements.
takeWhile :: (a  Bool)  [a]  [a]
takeWhile p [] = []
takeWhile p (x:xs)
| p x = x : takeWhile p xs
| otherwise = []
For example:
> takeWhile isAlpha "abc def"
"abc"

44. Dually, the function dropWhile removes elements while a predicate holds of all the elements.
dropWhile :: (a  Bool)  [a]  [a]
dropWhile p [] = []
dropWhile p (x:xs)
| p x = dropWhile p xs
| otherwise = x:xs
For example:
> dropWhile isSpace " abc"
"abc"