In the modern literature, "type theory" refers to a typed system based around lambda calculus. One influential system is Per Martin-Löf 's intuitionistic type theory, which was proposed as a …

As the main contribution, we devise a restricted type theory where identity types are derivable. We use the semantic intuition of chain complexes to introduce a type former called a null type, …

In JavaScript with Closure Compiler annotations, the bottom type is !Null (literally, a non-null member of the Null unit type).

description. Given a type A, and a type B(x) for every x : A, there is a type P x:A B(x), called the dependent pair type or -type (read \ Given a : A and b : B(a), there is a term (a; b) : P B(x).

Nov 20, 2020 · In a simple type theory (i.e. without dependent types), you could write the computation rule for the empty type as the following (note that this matches the rule in the …

Feb 8, 2006 · The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory and was introduced in “Appendix B: The …

In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave.

We discuss the Axiom of Choice, which is enormously useful in simplifying cardinal arithmetic, but which in type theory can be most naturally initially introduced as a formal device to simplify …

Aug 24, 2024 · In dependent type theory, given a type A A, a type T T is a A A -null type if the function. is an equivalence of types. More generally, given a type A A and a type family B(a) B …

We can approximate the axiomatic basis for constructive type theory as having fourteen basic types with an introduction and elimination rules for each, so with approximately twenty eight …