Level

This module is devoted to a number of tools useful for working with homotopy levels of universes.

Use level

The homotopy level of a definition is inferred automatically, but sometimes it is possible to prove that it has a smaller level. For example, we can define the following data types:

\data Empty
\data Dec (P : \Prop) | yes P | no (P -> Empty)

The type of Empty is inferred to \Prop, which is the right universe for the type. However, this is not as easy for the type Dec P: the type of Dec P is inferred to \Set0, whereas it can be proven that Dec P is (-1)-type. Dec can be placed in \Prop by writing the proof, that any two elements of this type are equal, in the \where block of Dec. The proof must be written in the corresponding function, starting with keywords \use \level instead of \func:

\data Empty
\data Dec (P : \Prop) | yes P | no (P -> Empty)
  \where
    \use \level isProp {P : \Prop} (d1 d2 : Dec P) : d1 = d2
      | yes x1, yes x2 => path (\lam i => yes (Path.inProp x1 x2 @ i))
      | yes x1, no e2 => \case e2 x1 \with {}
      | no e1, yes x2 => \case e1 x2 \with {}
      | no e1, no e2 => path (\lam i => no (\lam x => (\case e1 x \return e1 x = e2 x \with {}) @ i))

Functions \use \level can be specified for \data, \record, \class, and \func (for functions they work differently, see below) definitions. They must have a particular type. First parameters of such a function must be parameters of the data type (or the function) or (some) fields of the class. The rest of parameters together with the result type must prove that the data type (or the function, or the class) has some homotopy level. That is, it must prove ofHLevel (D p_1 … p_k) n for some constant n, where D is the data type (or the function, or the class), p_1, … p_k are its parameters (or fields), and ofHLevel is defined as follows:

\func \infix 2 ofHLevel_-1+ (A : \Type) (n : Nat) : \Type \elim n
  | 0 => \Pi (a a' : A) -> a = a'
  | suc n => \Pi (a a' : A) -> (a = a') ofHLevel_-1+ n

Level of a type

Sometimes we need to know that some type has a certain homotopy level. For example, the result type of a lemma or a property must be a proposition. If the type does not belong to the corresponding universe, but it can be proved that it has the correct homotopy level, the keyword \level can be used to convince the typechecker to accept the definition. This keywords can be specified in the result type of a function, a lemma, a field, or a case expression. Its first argument is the type and the second is the proof that it belongs to some homotopy level.

For example, if A is a type such that p : \Pi (x y : A) -> x = y, then a lemma that proves A can be defined as follows:

\lemma lem : \level A p => {?}

Similarly, a property of type A can be defined as follows:

\record R {
  \property p : \level A p
}

While defining a function or a case expression over a truncated type with values in A, some clauses can be omitted if A belongs to an appropriate universe. If it is not, but there is a proof that it has the required homotopy level, then the keyword \level can be used to convince the typechecker that some clauses can be omitted. For example, if Trunc is a propositional truncation with constructor inT : A -> Trunc A, A and B are types, g : A -> B is function, and p : \Pi (x y : B) -> x = y, then the function, extending g to Trunc A can be defined simply as follows:

\func f (t : Trunc A) : \level B p
  | inT a => g a

Similarly, the keyword \level can be used in case expressions:

\func f' (t : Trunc A) => \case t \return \level B p \with {
  | inT a => g a
}

Use level for functions

If a \use \level function defined for some function f, this does not change the definition of f at all. So, in this case, \use \level is just a syntactic sugar. If some function f is defined with a \use \level annotation, this does not change the type of f, but it will be treated as a type in a lower universe in situations described in Level of a type.

For example, we can prove that isProp is itslef a proposition and then define lemmas which prove that some type is a proposition:

\func isProp (A : \Type) => \Pi (a a' : A) -> a = a'
  \where \use \level proof (A : \Type) : isProp (isProp A) => {?} -- the proof is omitted

\lemma lem : isProp (\Sigma (n : Nat) (n = 0)) => {?}

Without \use \level annotation it would be necessary to specify the proof in the definition of lem.

Squashed data types

A data type is marked as squashed if the universe of this data type is less than the universe in which it should be. This happens when the data type is \truncated or when there is a \use \level annotation for it. To use pattern matching on a squashed data type, either the universe of the resulting type should be less than or equal to the universe of the data type or the pattern matching should be a part of either \sfunc, \lemma, \use \level, or \scase.