# Exploring Haskell: Defining Functions

Going from conditional expressions and guarded equations to pattern matching, lambda expressions, and operator sections.

## Conditional Expressions

As in most programming languages, Haskell supports conditional expression, which also can be used to define a function.

```
-- Absolute integer
abs :: Int -> Int
abs n = if n >= 0 then n else -n
-- Sign of integer
signum :: Int -> Int
signum n = if n < 0 then -1 else if n == 0 then 0 else 1
```

## Guarded Equations

Guarded equation is a preferred alternative to a conditional expression in Haskell.

```
-- Absolute integer
abs :: Int -> Int
abs n | n >= 0 = n
| otherwise = -n
-- Sign of integer
signum :: Int -> Int
signum n | n < 0 = -1
| n == 0 = 0
| otherwise = 1
-- When otherwise is unspecified the default value is otherwise = True
```

## Pattern Matching

Pattern matching is a simple way to define a function by matching a pattern with an expected result.

```
-- Boolean negation
not :: Bool -> Bool
not False = True
not True = False
-- Boolean AND (Naive)
(&&) :: Bool -> Bool -> Bool
True && True = True
True && False = False
False && True = False
False && False = False
-- Boolean AND (Compact)
(&&) :: Bool -> Bool -> Bool
True && True = True
_ && _ = False
-- _ is a wildcard to match any symbol
-- Boolean AND (Lazy)
(&&) :: Bool -> Bool -> Bool
True && b = b
False && _ = False
```

Patterns are matched in order of definition, left to right, top to bottom.

```
-- Will always return False
(&&) :: Bool -> Bool -> Bool
_ && _ = False
True && True = True
```

Patterns do not repeating arguments.

```
-- Conflicting definition of b
(&&) :: Bool -> Bool -> Bool
b && b = b
_ && _ = False
-- Correct way is to use a guarded equation
(&&) :: Bool -> Bool -> Bool
b && c | b == c = b
| otherwise = False
```

## 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]
-- is actually
1:(2:(3:(4:[])))
```

Function on a list can be defined using a `x:xs`

pattern.

```
-- Return the first element of a given list
head :: [a] -> a
head (x : _) = x
-- Return given list without the first element
tail :: [a] -> [a]
tail (_ : xs) = xs
```

`x:xs`

can only match non-empty lists.`x:xs`

pattern must be inside parenthesis because of the order of operations.

## Lambda Expressions

A function can be constructed without naming the function by using a lambda expression.
For example: `λx -> x + x`

.

The symbol `λ`

is the Greek letter *lambda* and in Haskell is denoted with a `\`

.

### Usage of Lambda Expressions

Give formal meaning to a curried function.

```
-- Without lambda expression
add :: Int -> Int -> Int
add x y = x + y
-- With lambda expression
add :: Int -> Int -> Int
add = \x -> (\y -> x + y)
```

Define a function that returns another function as a result.

```
-- Without lambda expression
const :: a -> b -> a
const x _ = x
-- With lambda expression
const :: a -> (b -> a)
const x = \_ -> x
```

Avoid naming a function that is used once.

```
-- Without lambda expression
odds :: Int -> [Int]
odds n = map f [0 .. n - 1] where f x = x * 2 + 1
-- With lambda expression
odds :: Int -> [Int]
odds n = map (\x -> x * 2 + 1) [0 .. n - 1]
```

## Sections

An operator written between its two arguments can be converted into a curried function written before its two arguments by using parenthesis.

```
1 + 2 -- 3
(+) 1 2 -- 3
(1+) 2 -- 3
(+2) 1 -- 3
```

In general if `+`

is an operator then functions of the form `(+)`

, `(x+)`

, `(+y)`

are called sections.

### Using Sections

Sections can be used to instead of functions:

`(+)`

is the addition function`\x -> (\y -> x + y)`

`(1+)`

is the successor function`\y -> 1 + y`

`(1/)`

is the reciprocation function`\y -> 1 / y`

`(*2)`

is the doubling function`\x -> x * 2`

`(/2)`

is the halving function`\x -> x / 2`

And that's that.