Dependent types: formal definition and examples

Dependent types have been gaining a lot of attention among programmers. Some
would say it’s “the next thing” in programming. Why is it gaining popularity? In
this post, I hope to show some of the advantages of dependent types and get you
started on programming in dependent types.

To understand dependent types, let’s take a step back and review what types are.

What are types?

From Programming in Martin-Löf’s Type Theory: “Intuitively, a type is a
collection of objects together with an equivalence relation. Examples of types
are the type of sets, the type of elements in a set, the type of propositions,
…the type of predicates over a given set.” Let’s look at some of these
examples in detail.

The type of sets

“To know that A is a set is to know how to form the canonical elements in the
set and under what conditions two canonical elements are equal.”

Canonical elements are evaluated expressions or values.

In a universe of small types, which contains all types except itself, the type
of sets is called Type (or sometimes Set). Elements of Type include the
type of natural numbers, the type of booleans, etc.

The type of propositions

What are propositions? “To know that A is a proposition is to know that A is
a set.” That is, propositions are types! This leads to an important
feature/advantage of dependent types: types can depend on propositions!

What does it mean for a proposition to be true? “To know that the proposition A is true is to have an element a in A.”

For example, from Idris’s documentation onePlusOne below is a proposition of the equality type, the only constructor it has is Refl, which stands for reflexivity.

onePlusOne : 1+1=2
onePlusOne = Refl

Thus, the only way to construct the onePlusOne type is via the Refl
constructor, which we can only use if the left and right hand sides of the
equality sign evaluate to the same thing.

What are dependent types?

A dependent type is a type whose definition depends on a value. The classic
example is Vector A n, which is a list of type A with a specified length
n. The Vector type depends on a natural number (the value).

In a dependently typed language, types are values. But in a language like
Haskell, there is a distinction between polymorphic and dependent types.
Vector A n is both polymorphic and dependent. It’s polymorphic because it can
take any type A. It’s dependent because its output type depends on a value
n.

A datatype definition for dependent types

Here I present a definition of datatype that allows for dependent types. It’s similar to Agda’s.

In a universe of small types, we declare a datatype D, of type ? → Type, as follows:

data D (p₁ : P₁) ... (pₙ : Pₙ) : ? → Type
c₁ : ?₁ → D p₁ ... pₙ t₁¹ ... tₘ¹
...
cₖ : ?ₖ → D p₁ ... pₙ t₁ᵏ ... tₘᵏ

That’s a lot of symbols! Let’s go through them in detail:

• D is the name of the datatype.
• The colon : is read as “is of type”. E.g., p₁ : P₁ should be read as p₁
  is of type P₁.
• D takes in two types of inputs: parameters and indices. Parameters are the same for all constructors, indices can vary from constructor to constructor.
• p₁ ... pₙ are parameters of D. They are declared after the name of the datatype.
• The indices of D are of types ?.
• c₁ ... cₖ are constructors of D. They take arguments ?ᵢ and have the return types D p₁ ...
pₙ t₁ⁱ ... tₘⁱ where t₁ⁱ ... tₘⁱ are the arguments of the indices of D.

The following sections I illustrate some example datatypes in this context.

The type of natural numbers

The datatype of natural numbers can be introduced by:

data Nat : Type
zero : Nat
suc : Nat → Nat

Nat is a datatype of type Type. It has no parameter (there is no p). It is not indexed (? is empty).

zero is a constructor (c₁) of Nat. It doesn’t take any argument (?₁ is empty). Its type is Nat.

suc is another constructor (c₂) of Nat. It takes a Nat as an argument (?₂ = Nat). Its return type is Nat.

The type family of lists

The datatype family of lists is parameterised by A, without being indexed:

data List (A : Type) : Type
nil : List A
cons : A → List A → List A

List is a datatype of type Type. It has one parameter, p₁ = A. The type of the parameter is Type, so P₁ = Type.

It is not indexed (? is empty).

nil is a constructor (c₁) of List. It doesn’t take any argument (?₁ is empty). Its type is List A.

cons is another constructor (c₂) of List. It takes two arguments, ?₂ = A, List A. Its return type is List A.

The type family of vectors

The datatype family of vectors is parameterised by A, indexed by the length
of the list:

data Vec (A :Type) : Nat → Type
vnil : Vec A zero
vcons : {n : Nat} → A → Vec A n → Vec A (suc n)

Vec is a datatype of type Nat → Type. It has one parameter, p₁ = A. The type of the parameter is Type, so P₁ = Type.

It is indexed by a Nat, so ? = Nat.

vnil is a constructor (c₁) of Vec. It doesn’t take any argument (?₁ is empty). Its type is Vec A zero, so t₁¹ = zero.

vcons is another constructor (c₂) of Vec. It takes 2 arguments, ?₂ = A, Vec A n (the type of n is specified by the implicit argument to be
Nat). Its return type is Vec A (suc n), so t₁² = suc n.

The type family of identity type

The family of identity type can be defined by

data Id (A : Type) (x : A) : A → Type
refl : Id A x x

Id is a datatype of type A → Type. It has two parameters, p₁ =
A, p₂ = x. The types of the parameters are Type and A, so P₁ =
Type, P₂ = A.

It is indexed by x of type A, so ? = A.

refl is a constructor (c₁) of Id. It doesn’t take any argument
(?₁ is empty). Its type is Id A x x, so t₁¹ = x.

The type of Even

Dependent types can also be used to describe predicates. For example the
predicate of even numbers, Even : Nat → Type, can be defined as follows:

data Even : Nat → Type where
even-zero : Even zero
even-plus2 : {n : Nat} → Even n → Even (suc (suc n))

Even is a datatype of type Nat → Type. It has no parameter.

It is indexed by a Nat, so ? = Nat.

even-zero is a constructor (c₁) of Even. It doesn’t take any argument
(?₁ is empty). Its type is Even zero, so t₁¹ = zero.

even-plus2 is another constructor (c₂) of Even. It takes an
argument Even n (the type of n is specified by the implicit argument to be
Nat) (?₂ = Even n). Its type is Even (suc (suc n)), so
t₁² = (suc (suc n)).

To construct Even, the index has to either be

• zero, in which case we can construct Even zero with even-zero,
• or suc (suc n) such that Even n can be constructed. Using that Even n
  and even-plus2, we can construct Even (suc (suc n)). Therefore, it has to be
  two = (suc (suc zero)) or (suc (suc two)) and so on.

We’ve learned that dependent types allow us to define more intricate types
using parameters and indices. They also allow us to construct predicates and proofs. In
the next post, I will show some more real world dependent type programming!