# Structural types¶

In addition to the primitive types covered so far, disco also has sum and product types which can be used to build up more complex structures out of simpler ones.

## Product types¶

The product of two types, written using an asterisk * or Unicode times symbol ×, is a type whose values are ordered pairs of values of the component types. Pairs are written using standard ordered pair notation.

pair1 : N * Q
pair1 = (3, -5/6)

pair2 : Z5 × Bool
pair2 = (17 + 22, (3,5) < (4,2))

pair3 : Bool * (Bool * Bool)
pair3 = (true, (false, true))

pair4 : Bool * Bool * Bool
pair4 = (true, false, true)


pair1 in the example above has type N * Q, that is, the type of pairs of a natural number and a rational number; it is defined to be the pair containing 3 and -5/6. pair2 has type Z5 × Bool (using the alternate syntax × in place of *), and contains two values: $$(17 + 22) \pmod{5}$$, and the result of asking whether $$(3,5) < (4,2)$$.

Disco> pair2
(4, true)


Pairs are compared lexicographically, which intuitively means that the first component is most important, the second component breaks ties in the first component, and so on. For example, $$(a,b) < (c,d)$$ if either $$a < c$$ (in which case $$b$$ and $$d$$ don’t matter) or if $$a = c$$ and $$b < d$$. This is why (3,5) < (4,2) evaluates to true. Of course, two pairs are equal exactly when their first elements are equal and their second elements are equal.

pair3 shows that pairs can be nested: it is a pair whose second component is also a pair. pair4 looks like an ordered triple, but in fact we can check that pair3 and pair4 are equal!

Disco> pair3 = pair4
true


Really, pair4 is just syntax sugar for pair3. In general:

• The type X * Y * Z is interpreted as X * (Y * Z).
• The tuple (x,y,z) is interpreted as (x,(y,z)).

This continues recursively, so, for example, A * B * C * D * E means A * (B * (C * (D * E))). Put another way, disco really only has pairs, but appears to support arbitrarily large tuples by encoding them as right-nested pairs.

If you want left-nested pairs you can use explicit parentheses: for example, (Bool * Bool) * Bool is not the same as Bool * Bool * Bool, and has values such as ((false, true), true).

## Sum types¶

If X and Y are types, their sum, written X + Y (or X ⊎ Y), is the disjoint union of X and Y. That is, values of type X + Y are either values of X or values of Y, along with a “tag” so that we know which it is. The possible tags are left and right (to indicate the type on the left or right of the +). For example:

sum1 : N + Bool
sum1 = left 3

sum2 : N + Bool
sum2 = right false

sum3 : N + N + N
sum3 = right (right 3)


sum1 and sum2 have the same type, namely, N + Bool; values of this type consist of either a natural number or a boolean. sum1 contains a natural number, tagged with left; sum2 contains a boolean tagged with right.

Notice that X + X is a different type than X, because we get two distinct copies of all the values in X, some tagged with left and some with right. This is why we call a sum type a disjoint union.

Iterated sum types, as in sum3, are handled in exactly the same way as iterated product types: N + N + N is really syntax sugar for N + (N + N). sum3 therefore begins with a right tag, to show that it contains a value of the right-hand type, namely, N + N; this value in turn consists of another right tag along with a value of type N. Other values of the same type N + N + N include right (left 6) and left 5.

## Unit and Void types¶

Disco has two other special built-in types which are rarely useful on their own, but often play an important role in describing other types.

• The type Unit has just a single value, called ().

Disco> :type ()
() : Unit


It is isomorphic to the type Z1.

• The type Void has no values. It is also isomorphic to Z0.

## Counting and enumerating types¶

For any type which has only a finite number of values, disco can count how many values there are, using the count operator, or list them using enumerate (we will learn more about lists later in the tutorial).

Disco> count ((Z2 * Z4) + Bool)
right 10
Disco> enumerate ((Z2 * Z4) + Bool)
[left (0, 0), left (0, 1), left (0, 2), left (0, 3),
left (1, 0), left (1, 1), left (1, 2), left (1, 3),
right false, right true]
Disco> enumerate (Bool * Bool * Bool)
[(false, false, false), (false, false, true), (false, true, false), (false, true, true),
(true, false, false), (true, false, true), (true, true, false), (true, true, true)]