What is dependent typing?

Consider this: in all decent programming languages you can write functions, e.g.

def f(arg) = result

Here, f takes a value arg and computes a value result. It is a function from values to values.

Now, some languages allow you to define polymorphic (aka generic) values:

def empty<T> = new List<T>()

Here, empty takes a type T and computes a value. It is a function from types to values.

Usually, you can also have generic type definitions:

type Matrix<T> = List<List<T>>

This definition takes a type and it returns a type. It can be viewed as a function from types to types.

So much for what ordinary languages offer. A language is called dependently typed if it also offers the 4th possibility, namely defining functions from values to types. Or in other words, parameterizing a type definition over a value:

type BoundedInt(n) = {i:Int | i<=n}

Some mainstream languages have some fake form of this that is not to be confused. E.g. in C++, templates can take values as parameters, but they have to be compile-time constants when applied. Not so in a truly dependently-typed language. For example, I could use the type above like this:

def min(i : Int, j : Int) : BoundedInt(j) =
  if i < j then i else j

Here, the function’s result type depends on the actual argument value j, thus the terminology.

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