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

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)
Disco> g(-5)
Disco> factorial(5 + 6)

“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:

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:

thrice : (N -> N) -> (N -> N)
thrice(f)(n) = f(f(f(n)))

Anonymous functions

The syntax for an anonymous function in disco consists of a lambda (either a backslash or an actual λ) followed by one or more bindings, a period, and an arbitrary disco expression (the body).

Each binding is a pattern, which could be a single variable name, or a more complex pattern. Note that patterns can also contain type annotations. There can be multiple bindings separated by commas, which creates a (curried) “multi-argument” function.

Here are a few examples to illustrate the possibilities:

Disco> thrice(\x. x*2)(1)
Disco> thrice(\z:Nat. z^2 + 2z + 1)(7)
Disco> (\(x,y). x + y) (3,2)
Disco> (\x:N, y:Q. x > y) 5 (9/2)

Let expressions

Let expressions are a mechanism for defining new variables for local use within an expression. For example, 3 + (let y = 2 in y + y) evaluates to 7: the expression y + y is evaluated in a context where y is defined to be 2, and the result is then added to 3. The simplest syntax for a let expression, as in this example, is let <variable> = <expression1> in <expression2>. The value of the let expression is the value of <expression2>, which may contain occurrences of the <variable>; any such occurrences will take on the value of <expression1>.

More generally:

  • A let may have multiple variables defined before in, separated by commas.
  • Each variable may optionally have a type annotation.
  • The definitions of later variables may refer to previously defined variables.
  • However, the definition of a variable in a let may not refer to itself; only top-level definitions may be recursive.

Here is a (somewhat contrived) example which demonstrates all these features:

f : Nat -> List(Nat)
f n =
  let x : Nat = n//2,
      y : Nat = x + 3,
      z : List(Nat) = [3,x,y]
  in  n :: z

An important thing to note is that a given definition in a let expression will only ever be evaluated (at most) once, even if the variable is used multiple times. let expressions are thus a way for the programmer to ensure that the result of some computation is shared. let x = e in f x x and f e e will always yield the same result, but the former might be more efficient, if e is expensive to calculate.

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; 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.


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.

Operator functions

Operators can be manipulated as functions using the ~ notation. The tilde goes wherever the argument to the operator would go. This can be used, for example, to pass an operator to a higher-order function.

Disco> :type ~+~
~+~ : ℕ × ℕ → ℕ

Disco> import list
Loading list.disco...
Disco> foldr(~+~,0,[1 .. 10])

Disco> -- factorial
Disco> :type ~!
~! : ℕ → ℕ

Disco> -- negation
Disco> :type -~
-~ : ℤ → ℤ