Editing Type theory (section)

====Identity type====
Following the notion of Curry-Howard Correspondence, the [[identity type]] is a type introduced to mirror [[Propositional logic|propositional equivalence]], as opposed to the [[Judgment (mathematical logic)|judgmental (syntactic) equivalence]] that type theory already provides.

An identity type requires two terms of the same type and is written with the symbol <math>=</math>. For example, if <math>x+1</math> and <math>1+x</math> are terms, then <math>x+1=1+x</math> is a possible type. Canonical terms are created with a reflexivity function, <math>\mathrm{refl}</math>. For a term <math>t</math>, the call <math>\mathrm{refl}\,t</math> returns the canonical term inhabiting the type <math>t=t</math>.

The complexities of equality in type theory make it an active research topic; [[homotopy type theory]] is a notable area of research that mainly deals with equality in type theory.