# Currying

Currying is the process of transforming a function that takes multiple arguments in a tuple as its argument, into a function that takes just a single argument and returns another function which accepts further arguments, one by one, that the original function would receive in the rest of that tuple.

```
f :: a -> (b -> c) -- which can also be written as f :: a -> b -> c
```

is the **curried** form of

```
g :: (a, b) -> c
```

You can convert these two types in either directions with the Prelude functions `curry`

and `uncurry`

.

```
f = curry g
g = uncurry f
```

Both forms are equally expressive. It holds

```
f x y = g (x,y) ,
```

however the curried form is usually more convenient because it allows partial application.

In Haskell, *all* functions are considered curried: That is, *all functions in Haskell take just one argument.*
This is mostly hidden in notation, and so may not be apparent to a new Haskeller. It can be said that arrows in the types notation associate *to the right*, so that `f :: a -> b -> c`

is really `f :: a -> (b -> c)`

. Functional application, correspondingly, associates *to the left*: `f x y`

is really `(f x) y`

, so the types fit.

As an illustration, let's take the function

```
div :: Int -> Int -> Int -- which is actually Int -> (Int -> Int)
```

which performs integer division. The expression `div 11 2`

unsurprisingly evaluates to `5`

.

But there's more going on here than immediately meets the untrained eye. It could be a two-part process.

On its own, `div 11`

is a function of type `Int -> Int`

. Then that resulting function can be applied to the value `2`

, so `(div 11) 2`

yields `5`

. Of course an optimizing compiler will probably handle that whole expression at once, but conceptually that's what's going on.

You'll notice that the notation for types reflects this: you can read `Int -> Int -> Int`

incorrectly as "takes two `Int`

s and returns an `Int`

", but what it's *really* saying is "takes an `Int`

and returns something of the type `Int -> Int`

" -- that is, it returns a function that takes an `Int`

and returns an `Int`

. (One can write the type as `Int x Int -> Int`

if you really mean the former -- but since all functions in Haskell are curried, that's not legal Haskell. Alternatively, using tuples, you can write `(Int, Int) -> Int`

, but keep in mind that the tuple constructor `(,)`

itself can be curried.)

Much of the time, currying can be ignored by the new programmer. The major advantage of considering all functions as curried is theoretical: formal proofs are easier when all functions are treated uniformly (one argument in, one result out). Having said that, there *are* Haskell idioms and techniques for which you need to understand currying.

See

- partial application
- Section of an infix operator
- Sometimes it's valuable to think about functions abstractly without specifically giving all their arguments: this is the Pointfree style.
- Sometimes half the work of the function can be done looking only at the first argument (but there really
*is*only one argument, remember?): see functional dispatch. - Conversion between curried and uncurried style allows composition of functions with multiple values

## Exercises

- Simplify
`curry id`

- Simplify
`uncurry const`

- Express
`snd`

using`curry`

or`uncurry`

and other basic Prelude functions and without lambdas - Write the function
`\(x,y) -> (y,x)`

without lambda and with only Prelude functions