Abstract
AbstractWe give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.
Funder
Air Force Office of Scientific Research
Publisher
Springer Science and Business Media LLC
Subject
Artificial Intelligence,Computational Theory and Mathematics,Software
Reference40 articles.
1. Aczel, P.: An introduction to inductive definitions. Stud. Logic Found. Math. 90, 739–782 (1977)
2. Altenkirch, T., Kaposi, A.: Type theory in type theory using quotient inductive types. ACM SIGPLAN Notices 51(1), 18–29 (2016)
3. Angiuli, C., Hou (Favonia), K.-B., Harper, R.: Cartesian cubical computational type theory: constructive reasoning with paths and equalities. In: Ghica, D., Jung, A. (eds.) CSL 2018 (2018). https://doi.org/10.4230/LIPIcs.CSL.2018.6
4. Annenkov, D., Capriotti, P., Kraus, N., Sattler, C.: Two-level type theory and applications. arXiv:1705.03307 (2019)
5. Bauer, A., Gilbert, G., Haselwarter, P.G., Pretnar, M., Stone, C.A.: Design and implementation of the andromeda proof assistant. In: TYPES’16 (2018). https://doi.org/10.4230/lipics.types.2016.5