Sort System

Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and it denotes the universe of small types. But for some applications, other sorts are needed. This page explains the need for additional sorts and describes all the sorts that are used by Agda.

Introduction to universes

Russell’s paradox implies that the collection of all sets is not itself a set. Namely, if there were such a set U, then one could form the subset A ⊆ U of all sets that do not contain themselves. Then we would have A ∈ A if and only if A ∉ A, a contradiction.

Likewise, Martin-Löf’s type theory had originally a rule Set : Set but Girard showed that it is inconsistent. This result is known as Girard’s paradox. Hence, not every Agda type is a Set. For example, we have

Bool : Set
Nat : Set

but not Set : Set. However, it is often convenient for Set to have a type of its own, and so in Agda, it is given the type Set₁:

Set : Set₁

In many ways, expressions of type Set₁ behave just like expressions of type Set; for example, they can be used as types of other things. However, the elements of Set₁ are potentially larger; when A : Set₁, then A is sometimes called a large set. In turn, we have

Set₁ : Set₂
Set₂ : Set₃

and so on. A type whose elements are types is called a sort or a universe; Agda provides an infinite number of universes Set, Set₁, Set₂, Set₃, …, each of which is an element of the next one. In fact, Set itself is just an abbreviation for Set₀. The subscript n is called the level of the universe Setₙ.

Note

You can also write Set1, Set2, etc., instead of Set₁, Set₂. To enter a subscript in the Emacs mode, type “\_1”.

Universe example

So why are universes useful? Because sometimes it is necessary to define, and prove theorems about, functions that operate not just on sets but on large sets. In fact, most Agda users sooner or later experience an error message where Agda complains that Set₁ != Set. These errors usually mean that a small set was used where a large one was expected, or vice versa.

For example, suppose you have defined the usual datatypes for lists and cartesian products:

data List (A : Set) : Set where
  [] : List A
  _::_ : A → List A → List A
data _×_ (A B : Set) : Set where
 _,_ : A → B → A × B
infixr 5 _::_
infixr 4 _,_
infixr 2 _×_

Now suppose you would like to define an operator Prod that inputs a list of n sets and takes their cartesian product, like this:

Prod (A :: B :: C :: []) = A × B × C

There is only one small problem with this definition. The type of Prod should be

Prod : List Set → Set

However, the definition of List A specified that A had to be a Set. Therefore, List Set is not a valid type. The solution is to define a special version of the List operator that works for large sets:

data List₁ (A : Set₁) : Set₁ where
  []   : List₁ A
  _::_ : A → List₁ A → List₁ A

With this, we can indeed define:

Prod : List₁ Set → Set
Prod []        = ⊤
Prod (A :: As) = A × Prod As

Universe polymorphism

To allow definitions of functions and datatypes that work for all possible universes Setᵢ, Agda provides a type Level of universe levels and level-polymorphic universes Set ℓ where ℓ : Level. For more information, see the page on universe levels.

Agda’s sort system

The implementation of Agda’s sort system is closely based on the theory of pure type systems. The full sort system of Agda consists of the following sorts:

• Setᵢ and its universe-polymorphic variant Set ℓ

• Propᵢ and its universe-polymorphic variant Prop ℓ

• Setωᵢ

Sorts Setᵢ and Set ℓ

As explained in the introduction, Agda has a hierarchy of sorts Setᵢ : Setᵢ₊₁, where i is any concrete natural number, i.e. 0, 1, 2, 3, … The sort Set is an abbreviation for Set₀.

You can also refer to these sorts with the alternative syntax Seti. That means that you can also write Set0, Set1, Set2, etc., instead of Set₀, Set₁, Set₂.

In addition, Agda supports the universe-polymorphic version Set ℓ where ℓ : Level (see universe levels).

Sorts Propᵢ and Prop ℓ

In addition to the hierarchy Setᵢ, Agda also supports a second hierarchy Propᵢ : Setᵢ₊₁ (or Propi) of proof-irrelevant propositions. Like Set, Prop also has a universe-polymorphic version Prop ℓ where ℓ : Level.

Sorts Setωᵢ

To assign a sort to types such as (ℓ : Level) → Set ℓ, Agda further supports an additional sort Setω that stands above all sorts Setᵢ.

Just as for Set and Prop, Setω is the lowest level at an infinite hierarchy Setωᵢ : Setωᵢ₊₁ where Setω = Setω₀. You can also refer to these sorts with the alternative syntax Setωi. That means that you can also write Setω0, Setω1, Setω2, etc., instead of Setω₀, Setω₁, Setω₂.

Now it is allowed, for instance, to declare a datatype in Setω. This means that Setω before the implementation of this hierarchy, Setω used to be a term, and there was no bigger sort than it in Agda. Now a type can be assigned to it, in this case, Setω₁.

However, unlike the standard hierarchy of universes Setᵢ, this second hierarchy Setωᵢ does not support universe polymorphism. This means that it is not possible to quantify over all Setωᵢ at once. For example, the expression ∀ {i} (A : Setω i) → A → A would not be a well-formed agda term. See the section on Setω on the page on universe levels for more information.

Concerning other applications, It should not be necessary to refer to these sorts during normal usage of Agda, but they might be useful for defining reflection-based macros.

Note

When --omega-in-omega is enabled, Setωᵢ is considered to be equal to Setω for all i (thus rendering Agda inconsistent).

Sort metavariables and unknown sorts

Under universe polymorphism, levels can be arbitrary terms, e.g., a level that contains free variables. Sometimes, we will have to check that some expression has a valid type without knowing what sort it has. For this reason, Agda’s internal representation of sorts implements a constructor (sort metavariable) representing an unknown sort. The constraint solver can compute these sort metavariables, just like it does when computing regular term metavariables.

However, the presence of sort metavariables also means that sorts of other types can sometimes not be computed directly. For this reason, Agda’s internal representation of sorts includes three additional constructors funSort, univSort, and piSort. These constructors compute to the proper sort once enough metavariables in their arguments have been solved.

Note

funSort, univSort and piSort are internal constructors that may be printed when evaluating a term. The user can not enter them, nor introduce them in agda code. All these constructors do not represent new sorts but instead, they compute to the right sort once their arguments are known.

funSort

The constructor funSort computes the sort of a function type even if the sort of the domain and the sort of the codomain are still unknown.

To understand how funSort works in general, let us assume the following scenario:

• sA and sB are two (possibly different) sorts.

• A : sA, meaning that A is a type that has sort sA.

• B : sB, meaning that B is a (possibly different) type that has sort sB.

Under these conditions, we can build the function type A → B : funSort sA sB. This type signature means that the function type A → B has a (possibly unknown) but well-defined sort funSort sA sB, specified in terms of the sorts of its domain and codomain.

If sA and sB happen to be known, then funSort sA sB can be computed to a sort value. We list below all the possible computations that funSort can perform:

funSort Setωᵢ    Setωⱼ    = Setωₖ            (where k = max(i,j))
funSort Setωᵢ    (Set b)  = Setωᵢ
funSort Setωᵢ    (Prop b) = Setωᵢ
funSort (Set a)  Setωⱼ    = Setωⱼ
funSort (Prop a) Setωⱼ    = Setωⱼ
funSort (Set a)  (Set b)  = Set (a ⊔ b)
funSort (Prop a) (Set b)  = Set (a ⊔ b)
funSort (Set a)  (Prop b) = Prop (a ⊔ b)
funSort (Prop a) (Prop b) = Prop (a ⊔ b)

Example: the sort of the function type ∀ {A} → A → A with normal form {A : _5} → A → A evaluates to funSort (univSort _5) (funSort _5 _5) where:

• _5 is a metavariable that represents the sort of A.

• funSort _5 _5 is the sort of A → A.

Note

funSort can admit just two arguments, so it will be iterated when the function type has multiple arguments. E.g. the function type ∀ {A} → A → A → A evaluates to funSort (univSort _5) (funSort _5 (funSort _5 _5))

univSort

univSort returns the successor sort of a given sort.

Example: the sort of the function type ∀ {A} → A with normal form {A : _5} → A evaluates to funSort (univSort _5) _5 where:

• univSort _5 is the sort where the sort of A lives, ie. the successor level of _5.

We list below all the possible computations that univSort can perform:

univSort (Set a)  = Set (lsuc a)
univSort (Prop a) = Set (lsuc a)
univSort Setωᵢ    = Setωᵢ₊₁

piSort

Similarly, piSort s1 s2 is a constructor that computes the sort of a Π-type given the sort s1 of its domain and the sort s2 of its codomain as arguments.

To understand how piSort works in general, we set the following scenario:

• sA and sB are two (possibly different) sorts.

• A : sA, meaning that A is a type that has sort sA.

• x : A, meaning that x has type A.

• B : sB, meaning that B is a type (possibly different than A) that has sort sB.

Under these conditions, we can build the dependent function type (x : A) → B : piSort sA (λ x → sB). This type signature means that the dependent function type (x : A) → B has a (possibly unknown) but well-defined sort piSort sA sB, specified in terms of the element x : A and the sorts of its domain and codomain.

We list below all the possible computations that piSort can perform:

piSort s1       (λ x → s2) = funSort s1 s2          (if x does not occur freely in s2)
piSort (Set ℓ)  (λ x → s2) = Setω                   (if x occurs rigidly in s2)
piSort (Prop ℓ) (λ x → s2) = Setω                   (if x occurs rigidly in s2)
piSort Setωᵢ    (λ x → s2) = Setωᵢ                  (if x occurs rigidly in s2)

With these rules, we can compute the sort of the function type ∀ {A} → ∀ {B} → B → A → B (or more explicitly, {A : _9} {B : _7} → B → A → B) to be piSort (univSort _9) (λ A → funSort (univSort _7) (funSort _7 (funSort _9 _7)))

More examples:

• piSort Level (λ l → Set l) evaluates to Setω

• piSort (Set l) (λ _ → Set l') evaluates to Set (l ⊔ l')

• univSort (Set l) evaluates to Set (lsuc l)

• piSort s (λ x -> Setωi) evaluates to funSort s Setωi