MLTT

Martin-Löf Type Theory (MLTT) is a dependent type theory developed by Per Martin-Löf as a constructive foundation for mathematics and a framework for dependently typed programming languages. In MLTT, types classify terms, and types themselves can depend on terms, enabling expressive constructs such as dependent function types (Pi types) and dependent pair types (Sigma types). The theory also includes inductively defined data types, notably natural numbers with a recursive principle, and identity types that encode equality as a type whose inhabitants are proofs.

MLTT adheres to the propositions-as-types correspondence, where logical propositions correspond to types and formal proofs to terms. This makes constructive reasoning explicit, as every proof provides a computational witness.

To tame paradoxes and create a scalable foundation, MLTT introduces universes: a hierarchy of types Type_0, Type_1, and so on, often with cumulativity. With normalization results, many formulations ensure consistency and decidability of type checking, at least for certain variants.

Equality in MLTT is intensional: the identity type does not identify all proofs of equality, and the principle of uniqueness of identity proofs may fail. Extensions such as Homotopy Type Theory add univalence and higher-dimensional equalities, broadening the theory beyond traditional MLTT.

MLTT has influenced modern proof assistants and programming languages with dependent types. Variants and implementations include Agda and Idris, and it informs the design of systems like Coq (which follows related ideas in the Calculus of Inductive Constructions).