Type theory

In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system.^[a] Type theory is the academic study of type systems.

Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that have been proposed as foundations are:

• Typed λ-calculus of Alonzo Church
• Intuitionistic type theory of Per Martin-Löf

Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions.
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