# Functions¶

The type of functions with input X and output Y is written X -> Y. Some basic examples of function definitions are shown below.

example/function.disco
f : N -> N
f(x) = x + 7

g : Z -> Bool
g(n) = (n - 3) > 7

factorial : N -> N
factorial(0) = 1
factorial(n) = n * factorial(n .- 1)

• The function f takes a natural number as input, and returns the natural number which is 7 greater. Notice that f is defined using the syntax f(x) = .... In fact, the basic syntax for function arguments is juxtaposition, just as in Haskell; the syntax f x = ... would work as well. Stylistically, however, f(x) = ... is to be preferred, since it matches standard mathematical notation.
• The function g takes an integer n as input, and returns a boolean indicating whether n - 3 is greater than 7. Note that this function cannot be given the type N -> Bool, since it uses subtraction.
• The recursive function factorial computes the factorial of its input. Top-level functions such as factorial are allowed to be recursive. Notice also that factorial is defined by two cases, which are matched in order from top to bottom, just as in Haskell.

Functions can be given inputs using the same syntax:

Disco> f(2^5)
39
Disco> g(-5)
false
Disco> factorial(5 + 6)
39916800


“Multi-argument functions” can be written as functions which take a product type as input. (This is again a stylistic choice: disco certainly supports curried functions as well. But in either case, disco fundamentally supports only one-argument functions.) For example:

example/multi-arg-functions.disco
gcd : N * N -> N
gcd(a,0) = a
gcd(a,b) = gcd(b, a mod b)

discrim : Q * Q * Q -> Q
discrim(a,b,c) = b^2 - 4*a*c

manhattan : (Q*Q) * (Q*Q) -> Q
manhattan ((x1,y1), (x2,y2)) = abs (x1-x2) + abs (y1-y2)


All of these examples are in fact pattern-matching on their arguments, although this is most noticeable with the last example, which decomposes its input into a pair of pairs and gives a name to each component.

Functions in disco are first-class, and can be provided as input to another function or output from a function, stored in data structures, etc. For example, here is how one could write a higher-order function to take a function on natural numbers and produce a new function which iterates the original function three times:

example/higher-order.disco
thrice : (N -> N) -> (N -> N)
thrice(f)(n) = f(f(f(n)))


## Anonymous functions¶

The syntax for an anonymous function in disco consists of three parts: one or more bindings, followed by a mapsto symbol, followed by an arbitrary disco expression.

• Each binding specifies the name of an input to the function. A binding can be either a simple variable name, or a parenthesized variable name with a type annotation (e.g. (x:Nat)). There can be multiple bindings separated by whitespace, which creates a (curried) “multi-argument” function.
• disco will accept any one of several syntaxes for the mapsto symbol: either ->, |->, or ↦.

Note

It’s quite possible this syntax might change. For example, we might want to disallow -> as a mapsto symbol, since that may cause confusion with the same symbol used as part of a type. Also, we might want to require “lambda” syntax before the binding (e.g. either a backslash or an actual lambda).

The current syntax was designed to mirror the syntax in most common mathematical practice (e.g. $$x \mapsto x^2 + 3$$), but it’s quite possible discrete math students will not be familiar with that notation anyway, in which case we might as well introduce them to the lambda calculus.

Currently, bindings cannot contain patterns, but in general we might want to allow this, for example, ((x,y) |-> x + y) : N*N -> N.

Here are a few examples of using anonymous functions as arguments to thrice:

Disco> thrice(x |-> x*2)(1)
8
Disco> thrice((z:Nat) ↦ z^2 + 2z + 1)(7)
17859076


TODO example of using multi-argument anonymous function

## Comparing functions¶

In certain cases, functions can be compared for equality, or even compared to see which is less or greater.

Disco> ((x:Bool) -> x) = ((x:Bool) -> not (not x))
true
Disco> ((x:Bool) -> x) = ((x:Bool) -> not x)
false


There is no magic involved, and it does not work by looking at the definitions of the functions. Simply put, two functions are equal if they give the same output for every input. So disco can only test two functions for equality if they have a finite input type, in which case it simply enumerates all possible values of the input type, and tests that the two functions give equal outputs for every input.

Functions are ordered by conceptually listing all their outputs ordered by inputs (that is, list the values of the input type in order from smallest to largest and apply the function to each) and then comparing these lists of outputs lexicographically. That is, if i is the smallest possible input value and f i < g i, then f < g. If f i = g i, then we move on to consider the second smallest input value, and so on.

## Disambiguating function application and multiplication¶

As previously mentioned, the fundamental syntax for applying a function to an argument is juxtaposition, that is, simply putting the function next to its argument (with a space in between if necessary).

However, disco also allows multiplication to be written in this way. How can it tell the difference? Given an expression of the form X Y (where X and Y may themselves be complex expressions), disco uses simple syntactic rules to distinguish between multiplication and function application. In particular, note that the types of X and Y do not enter into it at all (it would greatly complicate matters if parsing and typechecking had to be interleaved—even though this is what human mathematicians do in their heads; see the discussion below).

To decide whether X Y is function application or multiplication, disco looks only at the syntax of X; X Y is multiplication if and only if X is a multiplicative term, and function application otherwise. A multiplicative term is one that looks like either a natural number literal, or a unary or binary operation (possibly in parentheses). For example, 3, (-2), and (x + 5) are all multiplicative terms, so 3x, (-2)x, and (x + 5)x all get parsed as multiplication. On the other hand, an expression like (x y) is always parsed as function application, even if x and y both turn out to have numeric types types; a bare variable like x does not count as a multiplicative term. Likewise, (x y) z is parsed as function application, since (x y) is not a multiplicative term.

Note

You may enjoy reflecting on how a human mathematician does this disambiguation. In fact, they are doing something much more sophisticated than disco, implicitly using information about types and social conventions regarding variable names in addition to syntactic cues. For example, consider $$x(y + 3)$$ versus $$f(y + 3)$$. Most mathematicians would unconsciously interpret the first as multiplication and the second as function application, due to standard conventions about the use of variable names $$x$$ and $$f$$. On the other hand, in the sentence “Let $$x$$ be the function which doubles an integer, and consider $$v = x(y+3)$$“, any mathematician would have no trouble identifying this use of $$x(y+3)$$ as function application, although they might also rightly complain that $$x$$ is a strange choice for the name of a function.