core.typed supports polymorphic function types. They allow us to specify function types which are both general and accurate.

All

The primitive All constructor creates a polymorphic binder and scopes type variables in a type.

The identity function has a simple polymorphic type:

(All [x]
  [x -> x])

Read: for all types x, a function that takes an x and returns an x.

Polymorphic types are introduced with annotations, but where are they eliminated? We use local type inference to infer type variable types based on how they are used.

(identity :a)

In the above example, we infer x to be Keyword, and instantiate the polymorphic type as [Keyword -> Keyword].

Bounds

Type variables support upper and lower type bounds, which default to Any and Nothing respectively.

Equivalently, the type:

(All [x] ...)

is shorthand for:

(All [[x :> Nothing :< Any]] ...)

We use bounds to ensure a type variable can only be instantiated to a particular type.

The type of an identity function that only accepts Numbers can be written:

(All [[x :< Number]]
  [x -> x])

Bounds do not seem as useful in core.typed as languages like Java or Scala. Often, combinations of ordered function intersections and unions are more useful.

Bounds are also recursive: a bound can refer to the variable it's bounding. Type variables to the left of the type variable being bounded in the same binder are in scope in a bound.

Higher-kinded variables

Note: Experimental feature

A type variable can be of a higher-kind.

(def-alias AnyMonad 
  (TFn [[m :kind (TFn [[x :variance :covariant]] Any)]]
    '{:m-bind (All [x y]
                [(m x) [x -> (m y)] -> (m y)])
      :m-result (All [x]
                  [x -> (m x)])
      :m-zero (U (All [x] (m x)) Undefined)
      :m-plus (U (All [x]
                   [(m x) * -> (m x)])
                 Undefined)}))

In this type, x is a type function taking a type and returning a type. For those familiar with Haskell, x is of kind * -> *.

The type function is also covariant, which further ensures x is instantiated to a covariant type function.