Implementing a modal dependent type theory

Author:

Gratzer Daniel1,Sterling Jonathan2,Birkedal Lars1

Affiliation:

1. Aarhus University, Denmark

2. Carnegie Mellon University, USA

Abstract

Modalities are everywhere in programming and mathematics! Despite this, however, there are still significant technical challenges in formulating a core dependent type theory with modalities. We present a dependent type theory MLTT 🔒 supporting the connectives of standard Martin-Löf Type Theory as well as an S4 -style necessity operator. MLTT 🔒 supports a smooth interaction between modal and dependent types and provides a common basis for the use of modalities in programming and in synthetic mathematics. We design and prove the soundness and completeness of a type checking algorithm for MLTT 🔒 , using a novel extension of normalization by evaluation. We have also implemented our algorithm in a prototype proof assistant for MLTT 🔒 , demonstrating the ease of applying our techniques.

Funder

Villum Fonden

Air Force Office of Scientific Research

Natur og Univers, Det Frie Forskningsråd

Publisher

Association for Computing Machinery (ACM)

Subject

Safety, Risk, Reliability and Quality,Software

Reference59 articles.

1. A core calculus of dependency

2. Andreas Abel. 2013. Normalization by Evaluation: Dependent Types and Impredicativity. Andreas Abel. 2013. Normalization by Evaluation: Dependent Types and Impredicativity.

3. Normalization by Evaluation for Martin-Löf Type Theory with One Universe

4. A Modular Type-Checking Algorithm for Type Theory with Singleton Types and Proof Irrelevance

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