Editing Type theory (section)

==== Lambda terms ====
New function terms may be constructed using [[Lambda calculus#Definition|lambda expression]]s, and are called lambda terms. These terms are also defined inductively: a lambda term has the form <math>(\lambda v .t)</math>, where <math>v</math> is a formal variable and <math>t</math> is a term, and its type is notated <math>\sigma\to\tau</math>, where <math>\sigma</math> is the type of <math>v</math>, and <math>\tau</math> is the type of <math>t</math>.<ref name=":1" /> The following lambda term represents a function which doubles an input natural number.

<math>(\lambda x.\mathrm{add}\,x\,x): \mathsf{nat}\to\mathsf{nat}</math>

The variable is <math>   x</math> and (implicit from the lambda term's type) must have type <math>    \mathsf{nat} </math>. The term <math>   \mathrm{add}\,x\,x</math> has type <math>    \mathsf{nat} </math>, which is seen by applying the function application inference rule twice. Thus, the lambda term has type <math>\mathsf{nat}\to\mathsf{nat}</math>, which means it is a function taking a natural number as an [[Argument of a function|argument]] and returning a natural number.

A lambda term is often referred to{{efn|name= anon|1= In [[Julia (programming language)|Julia]], for example, a function with no name, but with two parameters in some tuple (x,y) can be denoted by say, <code>(x,y) -> x^5+y</code>, as an anonymous function.<ref name=balbaert > Balbaert,Ivo [https://www.oreilly.com/library/view/getting-started-with/9781783284795/ch03s03.html (2015) Getting Started with Julia]</ref>}} as an [[anonymous function]] because it lacks a name. The concept of anonymous functions appears in many programming languages.