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# Partial application

In computer science, partial application (or partial function application) refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function ${\displaystyle f\colon (X\times Y\times Z)\to N}$, we might fix (or 'bind') the first argument, producing a function of type ${\displaystyle {\text{partial}}(f)\colon (Y\times Z)\to N}$. Evaluation of this function might be represented as ${\displaystyle f_{partial}(2,3)}$. Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.

## Motivation

Intuitively, partial function application says "if you fix the first arguments of the function, you get a function of the remaining arguments". For example, if function div(x,y) = x/y, then div with the parameter x fixed at 1 is another function: div1(y) = div(1,y) = 1/y. This is the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1/y.

The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to plus_one. Partial application makes it easy to define these functions, for example by creating a function that represents the addition operator with 1 bound as its first argument.

## Implementations

In languages such as ML, Haskell and F#, functions are defined in curried form by default. Supplying fewer than the total number of arguments is referred to as partial application.

In languages with first-class functions, one can define curry, uncurry and papply to perform currying and partial application explicitly. This might incur a greater run-time overhead due to the creation of additional closures, while Haskell can use more efficient techniques.[1]

Scala implements optional partial application with placeholder, e.g. def add(x: Int, y: Int) = {x+y}; add(1, _: Int) returns an incrementing function. Scala also supports multiple parameter lists as currying, e.g. def add(x: Int)(y: Int) = {x+y}; add(1) _.

Clojure implements partial application using the partial function defined in its core library.[2]

The C++ standard library provides bind(function, args..) to return a function object that is the result of partial application of the given arguments to the given function. Alternatively, lambda expressions can be used:

int f(int a, int b);
auto f_partial = [](int a) { return f(a, 123); };
assert(f_partial(456) == f(456, 123) );


In Java, MethodHandle.bindTo partially applies a function to its first argument.[3] Alternatively, since Java 8, lambdas can be used:

public static <A, B, R> Function<B, R> partialApply(BiFunction<A, B, R> biFunc, A value) {
return b -> biFunc.apply(value, b);
}


In Raku, the assuming method creates a new function with fewer parameters.[4]

The Python standard library module functools includes the partial function, allowing positional and named argument bindings, returning a new function.[5]

In XQuery, an argument placeholder (?) is used for each non-fixed argument in a partial function application.[6]

## Definitions

In the simply-typed lambda calculus with function and product types (λ→,×) partial application, currying and uncurrying can be defined as

papply
(((a × b) → c) × a) → (bc) = λ(f, x). λy. f (x, y)
curry
((a × b) → c) → (a → (bc)) = λf. λx. λy. f (x, y)
uncurry
(a → (bc)) → ((a × b) → c) = λf. λ(x, y). f x y

Note that curry papply = curry.