For the Metatheory of Type Theory, Internal Sconing Is Enough
Paper in proceeding, 2023

Metatheorems about type theories are often proven by interpreting the syntax into models constructed using categorical gluing. We propose to use only sconing (gluing along a global section functor) instead of general gluing. The sconing is performed internally to a presheaf category, and we recover the original glued model by externalization. Our method relies on constructions involving two notions of models: first-order models (with explicit contexts) and higher-order models (without explicit contexts). Sconing turns a displayed higher-order model into a displayed first-order model. Using these, we derive specialized induction principles for the syntax of type theory. The input of such an induction principle is a boilerplate-free description of its motives and methods, not mentioning contexts. The output is a section with computation rules specified in the same internal language. We illustrate our framework by proofs of canonicity and normalization for type theory.

sconing

normalization

type theory

gluing

presheaves

canonicity

Author

Rafaël Bocquet

Eötvös Loránd University (ELTE)

Ambrus Kaposi

Eötvös Loránd University (ELTE)

Christian Sattler

Chalmers, Computer Science and Engineering (Chalmers), Computing Science

Leibniz International Proceedings in Informatics, LIPIcs

18688969 (ISSN)

Vol. 260 18
9783959772778 (ISBN)

8th International Conference on Formal Structures for Computation and Deduction, FSCD 2023
Rome, Italy,

Proof theory and higher categorical semantics of homotopy type theory

Swedish Research Council (VR) (2019-03765), 2020-01-01 -- 2023-12-31.

Subject Categories

Business Administration

Computer Science

DOI

10.4230/LIPIcs.FSCD.2023.18

More information

Latest update

11/15/2023