# Anonymous Functions

We already know that we can have values that are not bound to names. The integer 42, for example, can be entered at the toplevel without giving it a name:

# 42;;
- : int = 42


Or we can bind it to a name:

# let x = 42;;
val x : int = 42


Similarly, OCaml functions do not have to have names; they may be anonymous. For example, here is an anonymous function that increments its input: fun x -> x+1. Here, fun is a keyword indicating an anonymous function, x is the argument, and -> separates the argument from the body.

We now have two ways we could write an increment function:

let inc x = x + 1
let inc = fun x -> x+1


They are syntactically different but semantically equivalent. That is, even though they involve different keywords and put some identifiers in different places, they mean the same thing.

Anonymous functions are also called lambda expressions, a term that comes from the lambda calculus, which is a mathematical model of computation in the same sense that Turing machines are a model of computation. In the lambda calculus, fun x -> e would be written $\lambda x . e$. The $\lambda$ denotes an anonymous function.

It might seem a little mysterious right now why we would want functions that have no names. Don't worry; we'll see good uses for them later in the course. In particular, we will often create anonymous functions and pass them as input to other functions.

Syntax.

fun x1 ... xn -> e


Static semantics.

• If by assuming that x1:t1 and x2:t2 and ... and xn:tn, we can conclude that e:u, then fun x1 ... xn -> e : t1 -> t2 -> ... -> tn -> u.

Dynamic semantics.

An anonymous function is already a value. There is no computation to be performed.