Sociology

Polymer chemistry

Modalities

Mathematics

Social science

Biology

Algorithm

Mathematical proof

Normalization (sociology)

Recursion (computer science)

Chemistry

Ecology

Anthropology

Programming language

Proof theory

Geometry

Modal logic

Type (biology)

Computer science

Modal

Theoretical computer science

Type theory

We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof follows from a generalization of synthetic Tait computability—an abstract approach to gluing proofs—to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.

Normalization for Multimodal Type Theory