# Lists

Note

The API reference links in this article will take you to MSDN. The docs.microsoft.com API reference is not complete.

A list in F# is an ordered, immutable series of elements of the same type. To perform basic operations on lists, use the functions in the List module.

## Creating and Initializing Lists

You can define a list by explicitly listing out the elements, separated by semicolons and enclosed in square brackets, as shown in the following line of code.

let list123 = [ 1; 2; 3 ]


You can also put line breaks between elements, in which case the semicolons are optional. The latter syntax can result in more readable code when the element initialization expressions are longer, or when you want to include a comment for each element.

let list123 = [
1
2
3 ]


Normally, all list elements must be the same type. An exception is that a list in which the elements are specified to be a base type can have elements that are derived types. Thus the following is acceptable, because both Button and CheckBox derive from Control.

let myControlList : Control list = [ new Button(); new CheckBox() ]


You can also define list elements by using a range indicated by integers separated by the range operator (..), as shown in the following code.

let list1 = [ 1 .. 10 ]


An empty list is specified by a pair of square brackets with nothing in between them.

// An empty list.
let listEmpty = []


You can also use a sequence expression to create a list. See Sequence Expressions for more information. For example, the following code creates a list of squares of integers from 1 to 10.

let listOfSquares = [ for i in 1 .. 10 -> i*i ]


## Operators for Working with Lists

You can attach elements to a list by using the :: (cons) operator. If list1 is [2; 3; 4], the following code creates list2 as [100; 2; 3; 4].

let list2 = 100 :: list1


You can concatenate lists that have compatible types by using the @ operator, as in the following code. If list1 is [2; 3; 4] and list2 is [100; 2; 3; 4], this code creates list3 as [2; 3; 4; 100; 2; 3; 4].

let list3 = list1 @ list2


Functions for performing operations on lists are available in the List module.

Because lists in F# are immutable, any modifying operations generate new lists instead of modifying existing lists.

Lists in F# are implemented as singly linked lists, which means that operations that access only the head of the list are O(1), and element access is O(n).

## Properties

The list type supports the following properties:

Property Type Description
Head 'T The first element.
Empty 'T list A static property that returns an empty list of the appropriate type.
IsEmpty bool true if the list has no elements.
Item 'T The element at the specified index (zero-based).
Length int The number of elements.
Tail 'T list The list without the first element.

Following are some examples of using these properties.

let list1 = [ 1; 2; 3 ]

// Properties
printfn "list1.IsEmpty is %b" (list1.IsEmpty)
printfn "list1.Length is %d" (list1.Length)
printfn "list1.Item(1) is %d" (list1.Item(1))


## Using Lists

Programming with lists enables you to perform complex operations with a small amount of code. This section describes common operations on lists that are important to functional programming.

### Recursion with Lists

Lists are uniquely suited to recursive programming techniques. Consider an operation that must be performed on every element of a list. You can do this recursively by operating on the head of the list and then passing the tail of the list, which is a smaller list that consists of the original list without the first element, back again to the next level of recursion.

To write such a recursive function, you use the cons operator (::) in pattern matching, which enables you to separate the head of a list from the tail.

The following code example shows how to use pattern matching to implement a recursive function that performs operations on a list.

let rec sum list =
match list with
| [] -> 0


The previous code works well for small lists, but for larger lists, it could overflow the stack. The following code improves on this code by using an accumulator argument, a standard technique for working with recursive functions. The use of the accumulator argument makes the function tail recursive, which saves stack space.

let sum list =
let rec loop list acc =
match list with
| [] -> acc
loop list 0


The function RemoveAllMultiples is a recursive function that takes two lists. The first list contains the numbers whose multiples will be removed, and the second list is the list from which to remove the numbers. The code in the following example uses this recursive function to eliminate all the non-prime numbers from a list, leaving a list of prime numbers as the result.

let IsPrimeMultipleTest n x =
x = n || x % n <> 0

let rec RemoveAllMultiples listn listx =
match listn with
| [] -> listx

let GetPrimesUpTo n =
let max = int (sqrt (float n))
RemoveAllMultiples [ 2 .. max ] [ 1 .. n ]

printfn "Primes Up To %d:\n %A" 100 (GetPrimesUpTo 100)


The output is as follows:

Primes Up To 100:
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71; 73; 79; 83; 89; 97]


## Module Functions

The List module provides functions that access the elements of a list. The head element is the fastest and easiest to access. Use the property Head or the module function List.head. You can access the tail of a list by using the Tail property or the List.tail function. To find an element by index, use the List.nth function. List.nth traverses the list. Therefore, it is O(n). If your code uses List.nth frequently, you might want to consider using an array instead of a list. Element access in arrays is O(1).

### Boolean Operations on Lists

The List.isEmpty function determines whether a list has any elements.

The List.exists function applies a Boolean test to elements of a list and returns true if any element satisfies the test. List.exists2 is similar but operates on successive pairs of elements in two lists.

The following code demonstrates the use of List.exists.

// Use List.exists to determine whether there is an element of a list satisfies a given Boolean expression.
// containsNumber returns true if any of the elements of the supplied list match
// the supplied number.
let containsNumber number list = List.exists (fun elem -> elem = number) list
let list0to3 = [0 .. 3]
printfn "For list %A, contains zero is %b" list0to3 (containsNumber 0 list0to3)


The output is as follows:

For list [0; 1; 2; 3], contains zero is true


The following example demonstrates the use of List.exists2.

// Use List.exists2 to compare elements in two lists.
// isEqualElement returns true if any elements at the same position in two supplied
// lists match.
let isEqualElement list1 list2 = List.exists2 (fun elem1 elem2 -> elem1 = elem2) list1 list2
let list1to5 = [ 1 .. 5 ]
let list5to1 = [ 5 .. -1 .. 1 ]
if (isEqualElement list1to5 list5to1) then
printfn "Lists %A and %A have at least one equal element at the same position." list1to5 list5to1
else
printfn "Lists %A and %A do not have an equal element at the same position." list1to5 list5to1


The output is as follows:

Lists [1; 2; 3; 4; 5] and [5; 4; 3; 2; 1] have at least one equal element at the same position.


You can use List.forall if you want to test whether all the elements of a list meet a condition.

let isAllZeroes list = List.forall (fun elem -> elem = 0.0) list
printfn "%b" (isAllZeroes [0.0; 0.0])
printfn "%b" (isAllZeroes [0.0; 1.0])


The output is as follows:

true
false


Similarly, List.forall2 determines whether all elements in the corresponding positions in two lists satisfy a Boolean expression that involves each pair of elements.

let listEqual list1 list2 = List.forall2 (fun elem1 elem2 -> elem1 = elem2) list1 list2
printfn "%b" (listEqual [0; 1; 2] [0; 1; 2])
printfn "%b" (listEqual [0; 0; 0] [0; 1; 0])


The output is as follows:

true
false


### Sort Operations on Lists

The List.sort, List.sortBy, and List.sortWith functions sort lists. The sorting function determines which of these three functions to use. List.sort uses default generic comparison. Generic comparison uses global operators based on the generic compare function to compare values. It works efficiently with a wide variety of element types, such as simple numeric types, tuples, records, discriminated unions, lists, arrays, and any type that implements System.IComparable. For types that implement System.IComparable, generic comparison uses the System.IComparable.CompareTo() function. Generic comparison also works with strings, but uses a culture-independent sorting order. Generic comparison should not be used on unsupported types, such as function types. Also, the performance of the default generic comparison is best for small structured types; for larger structured types that need to be compared and sorted frequently, consider implementing System.IComparable and providing an efficient implementation of the System.IComparable.CompareTo() method.

List.sortBy takes a function that returns a value that is used as the sort criterion, and List.sortWith takes a comparison function as an argument. These latter two functions are useful when you are working with types that do not support comparison, or when the comparison requires more complex comparison semantics, as in the case of culture-aware strings.

The following example demonstrates the use of List.sort.

let sortedList1 = List.sort [1; 4; 8; -2; 5]
printfn "%A" sortedList1


The output is as follows:

[-2; 1; 4; 5; 8]


The following example demonstrates the use of List.sortBy.

let sortedList2 = List.sortBy (fun elem -> abs elem) [1; 4; 8; -2; 5]
printfn "%A" sortedList2


The output is as follows:

[1; -2; 4; 5; 8]


The next example demonstrates the use of List.sortWith. In this example, the custom comparison function compareWidgets is used to first compare one field of a custom type, and then another when the values of the first field are equal.

type Widget = { ID: int; Rev: int }

let compareWidgets widget1 widget2 =
if widget1.ID < widget2.ID then -1 else
if widget1.ID > widget2.ID then 1 else
if widget1.Rev < widget2.Rev then -1 else
if widget1.Rev > widget2.Rev then 1 else
0

let listToCompare = [
{ ID = 92; Rev = 1 }
{ ID = 110; Rev = 1 }
{ ID = 100; Rev = 5 }
{ ID = 100; Rev = 2 }
{ ID = 92; Rev = 1 }
]

let sortedWidgetList = List.sortWith compareWidgets listToCompare
printfn "%A" sortedWidgetList


The output is as follows:

[{ID = 92;
Rev = 1;}; {ID = 92;
Rev = 1;}; {ID = 100;
Rev = 2;}; {ID = 100;
Rev = 5;}; {ID = 110;
Rev = 1;}]


### Search Operations on Lists

Numerous search operations are supported for lists. The simplest, List.find, enables you to find the first element that matches a given condition.

The following code example demonstrates the use of List.find to find the first number that is divisible by 5 in a list.

let isDivisibleBy number elem = elem % number = 0
let result = List.find (isDivisibleBy 5) [ 1 .. 100 ]
printfn "%d " result


The result is 5.

If the elements must be transformed first, call List.pick, which takes a function that returns an option, and looks for the first option value that is Some(x). Instead of returning the element, List.pick returns the result x. If no matching element is found, List.pick throws System.Collections.Generic.KeyNotFoundException. The following code shows the use of List.pick.

let valuesList = [ ("a", 1); ("b", 2); ("c", 3) ]

let resultPick = List.pick (fun elem ->
match elem with
| (value, 2) -> Some value
| _ -> None) valuesList
printfn "%A" resultPick


The output is as follows:

"b"


Another group of search operations, List.tryFind and related functions, return an option value. The List.tryFind function returns the first element of a list that satisfies a condition if such an element exists, but the option value None if not. The variation List.tryFindIndex returns the index of the element, if one is found, rather than the element itself. These functions are illustrated in the following code.

let list1d = [1; 3; 7; 9; 11; 13; 15; 19; 22; 29; 36]
let isEven x = x % 2 = 0
match List.tryFind isEven list1d with
| Some value -> printfn "The first even value is %d." value
| None -> printfn "There is no even value in the list."

match List.tryFindIndex isEven list1d with
| Some value -> printfn "The first even value is at position %d." value
| None -> printfn "There is no even value in the list."


The output is as follows:

The first even value is 22.
The first even value is at position 8.


### Arithmetic Operations on Lists

Common arithmetic operations such as sum and average are built into the List module. To work with List.sum, the list element type must support the + operator and have a zero value. All built-in arithmetic types satisfy these conditions. To work with List.average, the element type must support division without a remainder, which excludes integral types but allows for floating point types. The List.sumBy and List.averageBy functions take a function as a parameter, and this function's results are used to calculate the values for the sum or average.

The following code demonstrates the use of List.sum, List.sumBy, and List.average.

// Compute the sum of the first 10 integers by using List.sum.
let sum1 = List.sum [1 .. 10]

// Compute the sum of the squares of the elements of a list by using List.sumBy.
let sum2 = List.sumBy (fun elem -> elem*elem) [1 .. 10]

// Compute the average of the elements of a list by using List.average.
let avg1 = List.average [0.0; 1.0; 1.0; 2.0]

printfn "%f" avg1


The output is 1.000000.

The following code shows the use of List.averageBy.

let avg2 = List.averageBy (fun elem -> float elem) [1 .. 10]
printfn "%f" avg2


The output is 5.5.

### Lists and Tuples

Lists that contain tuples can be manipulated by zip and unzip functions. These functions combine two lists of single values into one list of tuples or separate one list of tuples into two lists of single values. The simplest List.zip function takes two lists of single elements and produces a single list of tuple pairs. Another version, List.zip3, takes three lists of single elements and produces a single list of tuples that have three elements. The following code example demonstrates the use of List.zip.

let list1 = [ 1; 2; 3 ]
let list2 = [ -1; -2; -3 ]
let listZip = List.zip list1 list2
printfn "%A" listZip


The output is as follows:

[(1, -1); (2, -2); (3; -3)]


The following code example demonstrates the use of List.zip3.

let list3 = [ 0; 0; 0]
let listZip3 = List.zip3 list1 list2 list3
printfn "%A" listZip3


The output is as follows:

[(1, -1, 0); (2, -2, 0); (3, -3, 0)]


The corresponding unzip versions, List.unzip and List.unzip3, take lists of tuples and return lists in a tuple, where the first list contains all the elements that were first in each tuple, and the second list contains the second element of each tuple, and so on.

The following code example demonstrates the use of List.unzip.

let lists = List.unzip [(1,2); (3,4)]
printfn "%A" lists
printfn "%A %A" (fst lists) (snd lists)


The output is as follows:

([1; 3], [2; 4])
[1; 3] [2; 4]


The following code example demonstrates the use of List.unzip3.

let listsUnzip3 = List.unzip3 [(1,2,3); (4,5,6)]
printfn "%A" listsUnzip3


The output is as follows:

([1; 4], [2; 5], [3; 6])


### Operating on List Elements

F# supports a variety of operations on list elements. The simplest is List.iter, which enables you to call a function on every element of a list. Variations include List.iter2, which enables you to perform an operation on elements of two lists, List.iteri, which is like List.iter except that the index of each element is passed as an argument to the function that is called for each element, and List.iteri2, which is a combination of the functionality of List.iter2 and List.iteri. The following code example illustrates these functions.

let list1 = [1; 2; 3]
let list2 = [4; 5; 6]
List.iter (fun x -> printfn "List.iter: element is %d" x) list1
List.iteri(fun i x -> printfn "List.iteri: element %d is %d" i x) list1
List.iter2 (fun x y -> printfn "List.iter2: elements are %d %d" x y) list1 list2
List.iteri2 (fun i x y ->
printfn "List.iteri2: element %d of list1 is %d element %d of list2 is %d"
i x i y)
list1 list2


The output is as follows:

List.iter: element is 1
List.iter: element is 2
List.iter: element is 3
List.iteri: element 0 is 1
List.iteri: element 1 is 2
List.iteri: element 2 is 3
List.iter2: elements are 1 4
List.iter2: elements are 2 5
List.iter2: elements are 3 6
List.iteri2: element 0 of list1 is 1; element 0 of list2 is 4
List.iteri2: element 1 of list1 is 2; element 1 of list2 is 5
List.iteri2: element 2 of list1 is 3; element 2 of list2 is 6


Another frequently used function that transforms list elements is List.map, which enables you to apply a function to each element of a list and put all the results into a new list. List.map2 and List.map3 are variations that take multiple lists. You can also use List.mapi and List.mapi2, if, in addition to the element, the function needs to be passed the index of each element. The only difference between List.mapi2 and List.mapi is that List.mapi2 works with two lists. The following example illustrates List.map.

let list1 = [1; 2; 3]
let newList = List.map (fun x -> x + 1) list1
printfn "%A" newList


The output is as follows:

[2; 3; 4]


The following example shows the use of List.map2.

let list1 = [1; 2; 3]
let list2 = [4; 5; 6]
let sumList = List.map2 (fun x y -> x + y) list1 list2
printfn "%A" sumList


The output is as follows:

[5; 7; 9]


The following example shows the use of List.map3.

let newList2 = List.map3 (fun x y z -> x + y + z) list1 list2 [2; 3; 4]
printfn "%A" newList2


The output is as follows:

[7; 10; 13]


The following example shows the use of List.mapi.

let newListAddIndex = List.mapi (fun i x -> x + i) list1


The output is as follows:

[1; 3; 5]


The following example shows the use of List.mapi2.

let listAddTimesIndex = List.mapi2 (fun i x y -> (x + y) * i) list1 list2


The output is as follows:

[0; 7; 18]


List.collect is like List.map, except that each element produces a list and all these lists are concatenated into a final list. In the following code, each element of the list generates three numbers. These are all collected into one list.

let collectList = List.collect (fun x -> [for i in 1..3 -> x * i]) list1
printfn "%A" collectList


The output is as follows:

[1; 2; 3; 2; 4; 6; 3; 6; 9]


You can also use List.filter, which takes a Boolean condition and produces a new list that consists only of elements that satisfy the given condition.

let evenOnlyList = List.filter (fun x -> x % 2 = 0) [1; 2; 3; 4; 5; 6]


The resulting list is [2; 4; 6].

A combination of map and filter, List.choose enables you to transform and select elements at the same time. List.choose applies a function that returns an option to each element of a list, and returns a new list of the results for elements when the function returns the option value Some.

The following code demonstrates the use of List.choose to select capitalized words out of a list of words.

let listWords = [ "and"; "Rome"; "Bob"; "apple"; "zebra" ]
let isCapitalized (string1:string) = System.Char.IsUpper string1.[0]
let results = List.choose (fun elem ->
match elem with
| elem when isCapitalized elem -> Some(elem + "'s")
| _ -> None) listWords
printfn "%A" results


The output is as follows:

["Rome's"; "Bob's"]


### Operating on Multiple Lists

Lists can be joined together. To join two lists into one, use List.append. To join more than two lists, use List.concat.

let list1to10 = List.append [1; 2; 3] [4; 5; 6; 7; 8; 9; 10]
let listResult = List.concat [ [1; 2; 3]; [4; 5; 6]; [7; 8; 9] ]
List.iter (fun elem -> printf "%d " elem) list1to10
printfn ""
List.iter (fun elem -> printf "%d " elem) listResult


### Fold and Scan Operations

Some list operations involve interdependencies between all of the list elements. The fold and scan operations are like List.iter and List.map in that you invoke a function on each element, but these operations provide an additional parameter called the accumulator that carries information through the computation.

Use List.fold to perform a calculation on a list.

The following code example demonstrates the use of List.fold to perform various operations.

The list is traversed; the accumulator acc is a value that is passed along as the calculation proceeds. The first argument takes the accumulator and the list element, and returns the interim result of the calculation for that list element. The second argument is the initial value of the accumulator.

let sumList list = List.fold (fun acc elem -> acc + elem) 0 list
printfn "Sum of the elements of list %A is %d." [ 1 .. 3 ] (sumList [ 1 .. 3 ])

// The following example computes the average of a list.
let averageList list = (List.fold (fun acc elem -> acc + float elem) 0.0 list / float list.Length)

// The following example computes the standard deviation of a list.
// The standard deviation is computed by taking the square root of the
// sum of the variances, which are the differences between each value
// and the average.
let stdDevList list =
let avg = averageList list
sqrt (List.fold (fun acc elem -> acc + (float elem - avg) ** 2.0 ) 0.0 list / float list.Length)

let testList listTest =
printfn "List %A average: %f stddev: %f" listTest (averageList listTest) (stdDevList listTest)

testList [1; 1; 1]
testList [1; 2; 1]
testList [1; 2; 3]

// List.fold is the same as to List.iter when the accumulator is not used.
let printList list = List.fold (fun acc elem -> printfn "%A" elem) () list
printList [0.0; 1.0; 2.5; 5.1 ]

// The following example uses List.fold to reverse a list.
// The accumulator starts out as the empty list, and the function uses the cons operator
// to add each successive element to the head of the accumulator list, resulting in a
// reversed form of the list.
let reverseList list = List.fold (fun acc elem -> elem::acc) [] list
printfn "%A" (reverseList [1 .. 10])


The versions of these functions that have a digit in the function name operate on more than one list. For example, List.fold2 performs computations on two lists.

The following example demonstrates the use of List.fold2.

// Use List.fold2 to perform computations over two lists (of equal size) at the same time.
// Example: Sum the greater element at each list position.
let sumGreatest list1 list2 = List.fold2 (fun acc elem1 elem2 ->
acc + max elem1 elem2) 0 list1 list2

let sum = sumGreatest [1; 2; 3] [3; 2; 1]
printfn "The sum of the greater of each pair of elements in the two lists is %d." sum


List.fold and List.scan differ in that List.fold returns the final value of the extra parameter, but List.scan returns the list of the intermediate values (along with the final value) of the extra parameter.

Each of these functions includes a reverse variation, for example, List.foldBack, which differs in the order in which the list is traversed and the order of the arguments. Also, List.fold and List.foldBack have variations, List.fold2 and List.foldBack2, that take two lists of equal length. The function that executes on each element can use corresponding elements of both lists to perform some action. The element types of the two lists can be different, as in the following example, in which one list contains transaction amounts for a bank account, and the other list contains the type of transaction: deposit or withdrawal.

// Discriminated union type that encodes the transaction type.
type Transaction =
| Deposit
| Withdrawal

let transactionTypes = [Deposit; Deposit; Withdrawal]
let transactionAmounts = [100.00; 1000.00; 95.00 ]
let initialBalance = 200.00

// Use fold2 to perform a calculation on the list to update the account balance.
let endingBalance = List.fold2 (fun acc elem1 elem2 ->
match elem1 with
| Deposit -> acc + elem2
| Withdrawal -> acc - elem2)
initialBalance
transactionTypes
transactionAmounts
printfn "%f" endingBalance


For a calculation like summation, List.fold and List.foldBack have the same effect because the result does not depend on the order of traversal. In the following example, List.foldBack is used to add the elements in a list.

let sumListBack list = List.foldBack (fun acc elem -> acc + elem) list 0
printfn "%d" (sumListBack [1; 2; 3])

// For a calculation in which the order of traversal is important, fold and foldBack have different
// results. For example, replacing fold with foldBack in the listReverse function
// produces a function that copies the list, rather than reversing it.
let copyList list = List.foldBack (fun elem acc -> elem::acc) list []
printfn "%A" (copyList [1 .. 10])


The following example returns to the bank account example. This time a new transaction type is added: an interest calculation. The ending balance now depends on the order of transactions.

type Transaction2 =
| Deposit
| Withdrawal
| Interest

let transactionTypes2 = [Deposit; Deposit; Withdrawal; Interest]
let transactionAmounts2 = [100.00; 1000.00; 95.00; 0.05 / 12.0 ]
let initialBalance2 = 200.00

// Because fold2 processes the lists by starting at the head element,
// the interest is calculated last, on the balance of 1205.00.
let endingBalance2 = List.fold2 (fun acc elem1 elem2 ->
match elem1 with
| Deposit -> acc + elem2
| Withdrawal -> acc - elem2
| Interest -> acc * (1.0 + elem2))
initialBalance2
transactionTypes2
transactionAmounts2
printfn "%f" endingBalance2

// Because foldBack2 processes the lists by starting at end of the list,
// the interest is calculated first, on the balance of only 200.00.
let endingBalance3 = List.foldBack2 (fun elem1 elem2 acc ->
match elem1 with
| Deposit -> acc + elem2
| Withdrawal -> acc - elem2
| Interest -> acc * (1.0 + elem2))
transactionTypes2
transactionAmounts2
initialBalance2
printfn "%f" endingBalance3


The function List.reduce is somewhat like List.fold and List.scan, except that instead of passing around a separate accumulator, List.reduce takes a function that takes two arguments of the element type instead of just one, and one of those arguments acts as the accumulator, meaning that it stores the intermediate result of the computation. List.reduce starts by operating on the first two list elements, and then uses the result of the operation along with the next element. Because there is not a separate accumulator that has its own type, List.reduce can be used in place of List.fold only when the accumulator and the element type have the same type. The following code demonstrates the use of List.reduce. List.reduce throws an exception if the list provided has no elements.

In the following code, the first call to the lambda expression is given the arguments 2 and 4, and returns 6, and the next call is given the arguments 6 and 10, so the result is 16.

let sumAList list =
try
List.reduce (fun acc elem -> acc + elem) list
with
| :? System.ArgumentException as exc -> 0

let resultSum = sumAList [2; 4; 10]
printfn "%d " resultSum


### Converting Between Lists and Other Collection Types

The List module provides functions for converting to and from both sequences and arrays. To convert to or from a sequence, use List.toSeq or List.ofSeq. To convert to or from an array, use List.toArray or List.ofArray.