F# : Sequences |

**Sequences**, commonly called **sequence expressions**, are similar to lists: both data structures are used to represent an ordered collection of values. However, unlike lists, elements in a sequence are computed *as they are needed* (or "lazily"), rather than computed all at once. This gives sequences some interesting properties, such as the capacity to represent infinite data structures.

## Defining SequencesEdit

Sequences are defined using the syntax:

```
seq { expr }
```

Similar to lists, sequences can be constructed using ranges and comprehensions:

```
> seq { 1 .. 10 };; (* seq ranges *)
val it : seq<int> = seq [1; 2; 3; 4; ...]
> seq { 1 .. 2 .. 10 };; (* seq ranges *)
val it : seq<int> = seq [1; 3; 5; 7; ...]
> seq {10 .. -1 .. 0};; (* descending *)
val it : seq<int> = seq [10; 9; 8; 7; ...]
> seq { for a in 1 .. 10 do yield a, a*a, a*a*a };; (* seq comprehensions *)
val it : seq<int * int * int>
= seq [(1, 1, 1); (2, 4, 8); (3, 9, 27); (4, 16, 64); ...]
```

Sequences have an interesting property which sets them apart from lists: elements in the sequence are *lazily evaluated*, meaning that F# does not compute values in a sequence until the values are actually needed. This is in contrast to lists, where F# computes the value of all elements in a list on declaration. As a demonstration, compare the following:

```
> let intList =
[ for a in 1 .. 10 do
printfn "intList: %i" a
yield a ]
let intSeq =
seq { for a in 1 .. 10 do
printfn "intSeq: %i" a
yield a };;
val intList : int list
val intSeq : seq<int>
intList: 1
intList: 2
intList: 3
intList: 4
intList: 5
intList: 6
intList: 7
intList: 8
intList: 9
intList: 10
> Seq.nth 3 intSeq;;
intSeq: 1
intSeq: 2
intSeq: 3
intSeq: 4
val it : int = 4
> Seq.nth 7 intSeq;;
intSeq: 1
intSeq: 2
intSeq: 3
intSeq: 4
intSeq: 5
intSeq: 6
intSeq: 7
intSeq: 8
val it : int = 8
```

The list is created on declaration, but elements in the sequence are created as they are needed.

As a result, sequences are able to represent a data structure with an arbitrary number of elements:

```
> seq { 1I .. 1000000000000I };;
val it : seq<bigint> = seq [1I; 2I; 3I; 4I; ...]
```

The sequence above represents a list with one trillion elements in it. That does not mean the sequence actually contains one trillion elements, but it can *potentially* hold one trillion elements. By comparison, it would not be possible to create a list `[ 1I .. 1000000000000I ]`

since the .NET runtime would attempt to create all one trillion elements up front, which would certainly consume all of the available memory on a system before the operation completed.

Additionally, sequences can represent an infinite number of elements:

```
> let allEvens =
let rec loop x = seq { yield x; yield! loop (x + 2) }
loop 0;;
> for a in (Seq.take 5 allEvens) do
printfn "%i" a;;
0
2
4
6
8
val it : unit = ()
```

Notice the definition of `allEvens`

does not terminate. The function `Seq.take`

returns the first `n`

elements of elements of the sequence. If we attempted to loop through all of the elements, fsi would print indefinitely.

**Note:**sequences are implemented as state machines by the F# compiler. In reality, they manage state internally and hold only the last generated item in memory at a time. Memory usage is constant for creating and traversing sequences of any length.

## Iterating Through Sequences ManuallyEdit

The .NET Base Class Library (BCL) contains two interfaces in the `System.Collections.Generic`

namespace:

```
type IEnumerable<'a> =
interface
(* Returns an enumerator that iterates through a collection *)
member GetEnumerator<'a> : unit -> IEnumerator<'a>
end
type IEnumerator<'a> =
interface
(* Advances to the next element in the sequences. Returns true if
the enumerator was successfully advanced to the next element; false
if the enumerator has passed the end of the collection. *)
member MoveNext : unit -> bool
(* Gets the current element in the collection. *)
member Current : 'a
(* Sets the enumerator to its initial position, which is
before the first element in the collection. *)
member Reset : unit -> unit
end
```

The `seq`

type is defined as follows:

```
type seq<'a> = System.Collections.Generic.IEnumerable<'a>
```

As you can see, `seq`

is not a unique F# type, but rather another name for the built-in `System.Collections.Generic.IEnumerable`

interface. Since `seq`

/`IEnumerable`

is a native .NET type, it was designed to be used in a more imperative style, which can be demonstrated as follows:

```
open System
open System.Collections
let evens = seq { 0 .. 2 .. 10 } (* returns IEnumerable<int> *)
let main() =
let evensEnumerator = evens.GetEnumerator() (* returns IEnumerator<int> *)
while evensEnumerator.MoveNext() do
printfn "evensEnumerator.Current: %i" evensEnumerator.Current
Console.ReadKey(true) |> ignore
main()
```

This program outputs:

evensEnumerator.Current: 0 evensEnumerator.Current: 2 evensEnumerator.Current: 4 evensEnumerator.Current: 6 evensEnumerator.Current: 8 evensEnumerator.Current: 10

Behind the scenes, .NET converts every `for`

loop over a collection into an explicit while loop. In other words, the following two pieces of code compile down to the same bytecode:

```
let x = [1 .. 10]
for num in x do
printfn "%i" num
``` |
```
let x = [1 .. 10]
let enumerator = x.GetEnumerator()
while enumerator.MoveNext() do
let num = enumerator.Current
printfn "%i" num
``` |

All collections which can be used with the `for`

keyword implement the `IEnumerable<'a>`

interface, a concept which will be discussed later in this book.

## The `Seq`

ModuleEdit

Similar to the List modules, the `Seq`

module contains a number of useful functions for operating on sequences:

**val append : seq<'T> -> seq<'T> -> seq<'T>**

- Appends one sequence onto another sequence.

```
> let test = Seq.append (seq{1..3}) (seq{4..7});;
val it : seq<int> = seq [1; 2; 3; 4; ...]
```

**val choose : ('T -> 'U option) -> seq<'T> -> seq<'U>**

- Filters and maps a sequence to another sequence.

```
> let thisworks = seq { for nm in [ Some("James"); None; Some("John") ] |> Seq.choose id -> nm.Length }
val it : seq<int> = seq [5; 4]
```

**val distinct : seq<'T> -> seq<'T>**

- Returns a sequence that filters out duplicate entries.

```
> let dist = Seq.distinct (seq[1;2;2;6;3;2])
val it : seq<int> = seq [1; 2; 6; 3]
```

**val exists : ('T -> bool) -> seq<'T> -> bool**

- Determines if an element exists in a sequence.

```
> let equalsTwo x = x=2
> let exist = Seq.exists equalsTwo (seq{3..9})
val equalsTwo : int -> bool
val it : bool = false
```

**val filter : ('T -> bool) -> seq<'T> -> seq<'T>**

- Builds a new sequence consisting of elements filtered from the input sequence.

```
> Seq.filter (fun x-> x%2 = 0) (seq{0..9})
val it : seq<int> = seq [0; 2; 4; 6; ...]
```

**val fold : ('State -> 'T -> 'State) -> 'State -> seq<'T> -> 'State**

- Repeatedly applies a function to each element in the sequence from left to right.

```
> let sumSeq sequence1 = Seq.fold (fun acc elem -> acc + elem) 0 sequence1
Seq.init 10 (fun index -> index * index)
|> sumSeq
|> printfn "The sum of the elements is %d."
>
The sum of the elements is 285.
val sumSeq : seq<int> -> int
```

**Note:**sequences can only be read forward-only manner, so there is no corresponding`foldBack`

function as found in the List and Array modules.

**val initInfinite : (int -> 'T) -> seq<'T>**

- Generates a sequence consisting of an infinite number of elements.

```
> Seq.initInfinite (fun x -> x*x)
val it : seq<int> = seq [0; 1; 4; 9; ...]
```

**val map : ('T -> 'U) -> seq<'T> -> seq<'U>**

- Maps a sequence of type
`'a`

to type`'b`

.

```
> Seq.map (fun x->x*x+2) (seq[3;5;4;3])
val it : seq<int> = seq [11; 27; 18; 11]
```

**val item : int -> seq<'T> -> 'T**

- Returns the
*nth*value of a sequence.

```
> Seq.item 3 (seq {for n in 2..9 do yield n})
val it : int = 5
```

**val take : int -> seq<'T> -> seq<'T>**

- Returns a new sequence consisting of the first
*n*elements of the input sequence.

```
> Seq.take 3 (seq{1..6})
val it : seq<int> = seq [1; 2; 3]
```

**val takeWhile : ('T -> bool) -> seq<'T> -> seq<'T>**

- Return a sequence that, when iterated, yields elements of the underlying sequence while the given predicate returns
`true`

, and returns no further elements.

```
> let sequenciaMenorqDez = Seq.takeWhile (fun elem -> elem < 10) (seq {for i in 0..20 do yield i+1})
val sequenciaMenorqDez : seq<int>
> sequenciaMenorqDez;;
val it : seq<int> = seq [1; 2; 3; 4; ...]
```

**val unfold : ('State -> ('T * 'State) option) -> 'State seed -> seq<'T>**

- The opposite of
`fold`

: this function generates a sequence as long as the generator function returns`Some`

.

```
> let fibs = (0I, 1I) |> Seq.unfold (fun (a, b) -> Some(a, (b, a + b) ) );;
val fibs : seq<bigint>
> Seq.iter (fun x -> printf "%O " x) (Seq.take 20 fibs);;
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
```

The generator function in `unfold`

expects a return type of `('T * 'State) option`

. The first value of the tuple is inserted as an element into the sequence, the second value of the tuple is passed as the accumulator. The `fibs`

function is clever for its brevity, but it's hard to understand if you've never seen an unfold function. The following demonstrates `unfold`

in a more straightforward way:

```
> let test = 1 |> Seq.unfold (fun x ->
if x <= 5 then Some(sprintf "x: %i" x, x + 1)
else None );;
val test : seq<string>
> Seq.iter (fun x -> printfn "%s" x) test;;
x: 1
x: 2
x: 3
x: 4
x: 5
```

Often, it's preferable to generate sequences using `seq`

comprehensions rather than the `unfold`

.