# Type Definitions¶

The type keyword can be used to conveniently declare aliases for types. For example, consider the following function which takes a list of natural number triplets and returns the sum of all the triplets in the list:

example/tydefs.disco
sumTripletList : List (N * N * N) -> N
sumTripletList [] = 0
sumTripletList ((n1, n2, n3) :: rest) = (n1 + n2 + n3 + (sumTripletList rest))

Disco> sumTripletList [(1,2,3), (4,5,6)]
21


Let’s write the following type definition:

type NatTriple = N * N * N


The type NatTriple is defined as a 3-tuple containing three values of type N. Note that in Disco, all type names must begin with a capital letter. Now we can rewrite our type declaration for sumTripletList as follows:

sumTripletList : List(NatTriple) -> N


## Recursive type definitions¶

However, type definitions are in fact much more powerful. Disco has no special syntax for declaring algebraic data types as in Haskell, but unlike Haskell, type definitions in Disco can be recursive. Thus, we can build recursive algebraic data types directly. For example, we can define a type of binary trees with values of type N at nodes as follows:

example/tydefs.disco
type Tree = Unit + (N * Tree * Tree)


Here, we see that a Tree can either be a leaf, or a a triplet containing a natural number value of the root as the first element, and the left and right subtrees of type Tree as the second and third elements, respectively.

Given this definition of Tree, here is how we would write a function which takes a Tree and returns the sum of all its node values.

example/tydefs.disco
sumTree : Tree -> N
sumTree (left _) = 0
sumTree (right (n, l, r)) = n + sumTree(l) + sumTree(r)


## Parameterized type definitions¶

Type definitions can also be parameterized. For example, we can make the Tree type polymorphic:

example/tydefs-poly.disco
import list

type Tree(a) = Unit + (a * Tree(a) * Tree(a))

treeFold : b * (a * b * b -> b) * Tree(a) -> b
treeFold (z, f, left(unit)) = z
treeFold (z, f, right (a, l, r)) = f(a, treeFold(z,f,l), treeFold(z,f,r))

sumTree : Tree(Nat) -> Nat
sumTree(t) = treeFold(0, \(a,l,r). a+l+r, t)

flattenTree : Tree(a) -> List(a)
flattenTree(t) = treeFold([], \(a,l,r). append(l, append([a], r)), t)

Disco> :load example/tydefs-poly.disco
Disco> leaf = left(unit)
Disco> t = right (1, right (3, leaf, leaf), right (5, leaf, leaf)) : Tree N
Disco> sumTree(t)
9
Disco> flattenTree(t)
[3, 1, 5]


## Cyclic type definitions¶

There is only one restriction on recursive type definitions, namely, they are not allowed to be cyclic, i.e. unguardedly recursive. A type definition is cyclic if the following two conditions hold:

1. Repeated expansions of the type definition yield solely type definitions.
2. The same type definition is encountered twice during repeated expansion.

For example:

-- Foo is not allowed because it expands to itself.
type Foo = Foo
-- Bar is not allowed: it expands to Baz which expands to Bar.
type Bar = Baz
type Baz = Bar

-- Pair is OK (though rather useless) because it expands to a
-- top-level type former (product) which is not a type definition.
type Pair = Pair * Pair