The list is a fundamental data structure in functional programming.

A list having `x1`

, …, `xn`

as elements is written `List(x1, …, xn)`

:

```
val fruit = List("apples", "oranges", "pears")
val nums = List(1, 2, 3, 4)
val diag3 = List(List(1, 0, 0), List(0, 1, 0), List(0, 0, 1))
val empty = List()
```

- Lists are immutable --- the elements of a list cannot be changed,
- Lists are recursive (as you will see in the next subsection),
- Lists are
*homogeneous*: the elements of a list must all have the same type.

The type of a list with elements of type `T`

is written `List[T]`

:

```
val fruit: List[String] = List("apples", "oranges", "pears")
val nums: List[Int] = List(1, 2, 3, 4)
val diag3: List[List[Int]] = List(List(1, 0, 0), List(0, 1, 0), List(0, 0, 1))
val empty: List[Nothing] = List()
```

Actually, all lists are constructed from:

- the empty list
`Nil`

, and - the construction operation
`::`

(pronounced*cons*):`x :: xs`

gives a new list with the first element`x`

, followed by the elements of`xs`

(which is a list itself).

For example:

```
val fruit = "apples" :: ("oranges" :: ("pears" :: Nil))
val nums = 1 :: (2 :: (3 :: (4 :: Nil)))
val empty = Nil
```

Convention: Operators ending in “`:`

” associate to the right.

`A :: B :: C`

is interpreted as `A :: (B :: C)`

.

We can thus omit the parentheses in the definition above.

`val nums = 1 :: 2 :: 3 :: 4 :: Nil`

Operators ending in “`:`

” are also different in the fact that they are seen as method calls of
the *right-hand* operand.

So the expression above is equivalent to:

`val nums = Nil.::(4).::(3).::(2).::(1)`

It is possible to decompose lists with pattern matching:

`Nil`

: the`Nil`

constant,`p :: ps`

: A pattern that matches a list with a`head`

matching`p`

and a`tail`

matching`ps`

.

```
nums match {
// Lists of `Int` that starts with `1` and then `2`
case 1 :: 2 :: xs => …
// Lists of length 1
case x :: Nil => …
// Same as `x :: Nil`
case List(x) => …
// The empty list, same as `Nil`
case List() =>
// A list that contains as only element another list that starts with `2`
case List(2 :: xs) => …
}
```

Suppose we want to sort a list of numbers in ascending order:

- One way to sort the list
`List(7, 3, 9, 2)`

is to sort the tail`List(3, 9, 2)`

to obtain`List(2, 3, 9)`

. - The next step is to insert the head
`7`

in the right place to obtain the result`List(2, 3, 7, 9)`

.

This idea describes *Insertion Sort*:

```
def insertionSort(xs: List[Int]): List[Int] = xs match {
case List() => List()
case y :: ys => insert(y, insertionSort(ys))
}
```

Complete the definition insertion sort by filling in the blanks in the definition below:

```
val cond: (Int, Int) => Boolean = res0
def insert(x: Int, xs: List[Int]): List[Int] =
xs match {
case List() => x :: res1
case y :: ys =>
if (cond(x, y)) x :: y :: ys
else y :: insert(x, ys)
}
insert(2, 1 :: 3 :: Nil) shouldBe (1 :: 2 :: 3 :: Nil)
insert(1, 2 :: 3 :: Nil) shouldBe (1 :: 2 :: 3 :: Nil)
insert(3, 1 :: 2 :: Nil) shouldBe (1 :: 2 :: 3 :: Nil)
```

Transform the elements of a list using `map`

:

`List(1, 2, 3).map(x => x + 1) == List(2, 3, 4)`

Filter elements using `filter`

:

`List(1, 2, 3).filter(x => x % 2 == 0) == List(2)`

Transform each element of a list into a list and flatten the
results into a single list using `flatMap`

:

```
val xs =
List(1, 2, 3).flatMap { x =>
List(x, 2 * x, 3 * x)
}
xs == List(1, 2, 3, 2, 4, 6, 3, 6, 9)
```

We represent an optional value of type `A`

with the type `Option[A]`

.
This is useful to implement, for instance, partially defined
functions:

```
// The `sqrt` function is not defined for negative values
def sqrt(x: Double): Option[Double] = …
```

An `Option[A]`

can either be `None`

(if there is no value) or `Some[A]`

(if there is a value):

```
def sqrt(x: Double): Option[Double] =
if (x < 0) None else Some(…)
```

It is possible to decompose options with pattern matching:

```
def foo(x: Double): String =
sqrt(x) match {
case None => "no result"
case Some(y) => y.toString
}
```

Transform an optional value with `map`

:

```
Some(1).map(x => x + 1) shouldBe Some(2)
None.map((x: Int) => x + 1) shouldBe None
res0.map(x => x + 1) shouldBe res1
```

Filter values with `filter`

:

```
Some(1).filter(x => x % 2 == 0) shouldBe None
Some(2).filter(x => x % 2 == 0) shouldBe Some(2)
res0.filter(x => x % 2 == 0) shouldBe res1
```

Use `flatMap`

to transform a successful value into an optional value:

```
Some(1).flatMap(x => Some(x + 1)) shouldBe Some(2)
None.flatMap((x: Int) => Some(x + 1)) shouldBe None
res0.flatMap(x => Some(x + 1)) shouldBe res1
```

This subsection introduces types that are useful to handle failures.

`Try[A]`

represents a computation that attempted to return an `A`

. It can
either be:

- a
`Success[A]`

, - or a
`Failure`

.

The key difference between `None`

and `Failure`

s is that the latter provide
the reason of the failure:

```
def sqrt(x: Double): Try[Double] =
if (x < 0) Failure(new IllegalArgumentException("x must be positive"))
else Success(…)
```

`Try[A]`

values Like options and lists, `Try[A]`

is an algebraic data type, so it can
be decomposed using pattern matching.

`Try[A]`

also have `map`

, `filter`

and `flatMap`

. They behave the same
as with `Option[A]`

, excepted that any exception that is thrown
during their execution is converted into a `Failure`

.

`Either`

can also be useful to handle failures. Basically, the type
`Either[A, B]`

represents a value that can either be of type `A`

or
of type `B`

. It can be decomposed in two cases: `Left`

or `Right`

.

You can use one case to represent the failure and the other to represent
the success. One difference with `Try`

is that you can choose another
type than `Throwable`

to represent the exception. Another difference
is that exceptions that occur when transforming `Either`

values are
not converted into failures.

```
def sqrt(x: Double): Either[String, Double] =
if (x < 0) Left("x must be positive")
else Right(…)
```

`Either[A, B]`

Values Since Scala 2.12, `Either`

has `map`

and `flatMap`

. These methods
transform the `Right`

case only. We say that `Either`

is “right biased”:

```
Right(1).map((x: Int) => x + 1) shouldBe Right(2)
Left("foo").map((x: Int) => x + 1) shouldBe Left("foo")
```

`Either`

also has a `filterOrElse`

method that turns a `Right`

value
into a `Left`

value if it does not satisfy a given predicate:

`Right(1).filterOrElse(x => x % 2 == 0, "Odd value") shouldBe Left("Odd value")`

However, prior to Scala 2.12, `Either`

was “unbiased”. You had to explicitly
specify which “side” (`Left`

or `Right`

) you wanted to `map`

:

```
def triple(x: Int): Int = 3 * x
def tripleEither(x: Either[String, Int]): Either[String, Int] =
x.right.map(triple)
tripleEither(Right(1)) shouldBe res0
tripleEither(Left("not a number")) shouldBe res1
```