Type theory

In mathematical logic, and theoretical computer science, type theory is the study of formal systems that classify expressions or mathematical objects by their types. Roughly speaking, a type plays a similar role to that played by a data type in programming: it specifies what kind of thing an expression is and how it may be used. Type theories are used in the study of programming languages (type systems), formal logic, and the formalization of mathematics. Some type theories have been proposed as alternatives to set theory as a foundation of mathematics. Examples include Alonzo Church's simple theory of types and Per Martin-Löf's intuitionistic type theory. Many proof assistants are based on type theory. For example, the underlying formal language of Rocq (formerly Coq) is the calculus of inductive constructions, while Lean is based on dependent type theory.