Editing Type theory (section)

==== Dependent products and sums ====
Two common [[Dependent type|type dependencies]], dependent product and dependent sum types, allow for the theory to encode [[BHK interpretation|BHK intuitionistic logic]] by acting as equivalents to [[Quantification (logic)|universal and existential quantification]]; this is formalized by [[Curry–Howard correspondence|Curry–Howard Correspondence]].<ref name=":2" /> As they also connect to [[Cartesian product|products]] and [[Disjoint union|sums]] in [[set theory]], they are often written with the symbols <math>\Pi</math> and <math>\Sigma</math>, respectively.

Sum types are seen in [[Dependent type|dependent pairs]], where the second type depends on the value of the first term. This arises naturally in computer science where functions may return different types of outputs based on the input. For example, the Boolean type is usually defined with an eliminator function <math>\mathrm{if}</math>, which takes three arguments and behaves as follows.
* <math>\mathrm{if}\,\mathrm{true}\,x\,y</math> returns <math>x</math>, and
* <math>\mathrm{if}\,\mathrm{false}\,x\,y</math> returns <math>y</math>.
Ordinary definitions of <math>\mathrm{if}</math> require <math>x</math> and <math>y</math> to have the same type. If the type theory allows for dependent types, then it is possible to define a dependent type <math>x:\mathsf{bool}\,\vdash\,\mathrm{TF}\,x:U\to U\to U</math> such that

* <math>\mathrm{TF}\,\mathrm{true}\,\sigma\,\tau</math> returns <math>\sigma</math>, and
* <math>\mathrm{TF}\,\mathrm{false}\,\sigma\,\tau</math> returns <math>\tau</math>.

The type of <math>\mathrm{if}</math> may then be written as <math>\forall\,\sigma\,\tau.\Pi_{x:\mathsf{bool}}.\sigma\to\tau\to\mathrm{TF}\,x\,\sigma\,\tau</math>.

{{anchor|Equality types}}