Editing Type theory (section)

====Function terms====
In addition to atomic terms, most modern type theories also allow for [[Function (mathematics)|functions]]. Function types introduce an arrow symbol, and are [[Recursive definition|defined inductively]]: If <math>\sigma</math> and <math>\tau</math> are types, then the notation <math>\sigma\to\tau</math> is the type of a function which takes a [[parameter]] of type <math>\sigma</math> and returns a term of type <math>\tau</math>. Types of this form are known as [[Simply typed lambda calculus|''simple'' types]].<ref name=":1" />

Some terms may be declared directly as having a simple type, such as the following term, <math>\mathrm{add}</math>, which takes in two natural numbers in sequence and returns one natural number.

<math>\mathrm{add}:\mathsf{nat}\to (\mathsf{nat}\to\mathsf{nat})</math>

Strictly speaking, a simple type only allows for one input and one output, so a more faithful reading of the above type is that <math>\mathrm{add}</math> is a function which takes in a natural number and returns a function of the form <math>\mathsf{nat}\to\mathsf{nat}</math>. The parentheses clarify that <math>\mathrm{add}</math> does not have the type <math>(\mathsf{nat}\to \mathsf{nat})\to\mathsf{nat}</math>, which would be a function which takes in a function of natural numbers and returns a natural number. The convention is that the arrow is [[Operator associativity|right associative]], so the parentheses may be dropped from <math>\mathrm{add}</math>'s type.<ref name=":1" />