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For the Metatheory of Type Theory, Internal Sconing Is Enough

Authors Rafaël Bocquet , Ambrus Kaposi , Christian Sattler

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Author Details

Rafaël Bocquet
  • Eötvös Loránd University, Budapest, Hungary
Ambrus Kaposi
  • Eötvös Loránd University, Budapest, Hungary
Christian Sattler
  • Chalmers University of Technology, Gothenburg, Sweden

Funding

  • Kaposi, Ambrus: Supported by the Bolyai Fellowship of the Hungarian Academy of Sciences and by the "Application Domain Specific Highly Reliable IT Solutions" project of the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 funding scheme.

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Rafaël Bocquet, Ambrus Kaposi, and Christian Sattler. For the Metatheory of Type Theory, Internal Sconing Is Enough. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSCD.2023.18

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
Keywords
  • type theory
  • presheaves
  • canonicity
  • normalization
  • sconing
  • gluing

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