### Currying

Recall that in lambda calculus, a function can have more than one input, each preceded by a ** λ** symbol. Another way of thinking about more than one input is

**currying**.

**Currying a function of two inputs turns that function into a function with one input by passing one of the inputs into it.**In other words, currying turns

*f(x,y*) to

*g(y)*in which

*g*is

*f*with

*x*passed into it. And

*g*only takes one input,

*y*. For example:

f(x,y) = x + y ; if x = 3 then f(3,y) = 3 + y

Similarly in lambda calculus:

λx.λy.(x+y) 3 y = λy.(3+y) y

One can curry recursively, and turn a function of any number of input to a function of that number of input minus one. For example:

λx.λy.λz.(x+y+z) 3 4 5 = λy.λz.(3+y+z) 4 5 = λz.(3+4+z) 5 = (3+4+5)

Line 1 is a function that takes in 3 inputs (*x,y,z*). Line 2 is a function that takes in 2 inputs (*y,z*). Line 3 is a function that takes in 1 input (*z*).

### Currying in OCaml

Currying is also applied in popular functional languages including Haskell and OCaml. For example:

let fun1 x y z = x + y + z;; val fun1 : int -> int -> int -> int = <fun>

In line 1 I defined a function in OCaml that takes inputs *x,y,z*. Line 2 is the OCaml output, displaying the type signature of my function named *fun1*. How do we read it?

The first 3 *int* are the input types of *x,y,z*. The last *int* is the type of the output of *fun1*. You can see that OCaml treats *fun1* as a function that takes and passes in *x* (an *int*), and returns a function (with *x* passed in) that takes and passes in *y*, which returns a function (with *x* and *y* passed in) that takes in *z*. Lastly, the function (with all *x,y,z* passed in) returns an *int*.

Let’s take a look at another example:

let is_larger_than fun1 y = fun1 y > 100;; val is_larger_than : ('a -> int) -> 'a -> bool = <fun>

The first input type is** (‘a -> int)**. No currying is involved here. It is

**a function**that takes one input of a generic type

**, and returns an**

*(‘a)***. OCaml treats**

*int**is_larger_than*as a function that takes in

*fun1*, and returns a function (with

*fun1*passed in) that takes and passes in

*(‘a)*. The output of

*is_larger_than*is a

*bool*. If

*fun1*were to take more than one input, OCaml curries it:

let is_larger_than fun1 x y z = fun1 x y z > 100;; val is_larger_than : ('a -> 'b -> 'c -> int) -> 'a -> 'b -> 'c -> bool = <fun>

Again, the first input is a function. *fun1* takes in 3 inputs *x,y,z*. OCaml curries it, similar to above. OCaml treats *is_larger_than* as a function that takes and passes in *fun1*, and returns a function (with *fun1* passed in) that takes in *(‘a)*, and so on.

Currying applies in many other instances too. See this article.