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A reasonably gradual type theory

Authors:

Kenji Maillard,

Meven Lennon-Bertrand,

Nicolas Tabareau,

Éric TanterAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 6, Issue ICFP

Article No.: 124, Pages 931 - 959

https://doi.org/10.1145/3547655

Published: 31 August 2022 Publication History

Abstract

Gradualizing the Calculus of Inductive Constructions (CIC) involves dealing with subtle tensions between normalization, graduality, and conservativity with respect to CIC. Recently, GCIC has been proposed as a parametrized gradual type theory that admits three variants, each sacrificing one of these properties. For devising a gradual proof assistant based on CIC, normalization and conservativity with respect to CIC are key, but the tension with graduality needs to be addressed. Additionally, several challenges remain: (1) The presence of two wildcard terms at any type---the error and unknown terms---enables trivial proofs of any theorem, jeopardizing the use of a gradual type theory in a proof assistant; (2) Supporting general indexed inductive families, most prominently equality, is an open problem; (3) Theoretical accounts of gradual typing and graduality so far do not support handling type mismatches detected during reduction; (4) Precision and graduality are external notions not amenable to reasoning within a gradual type theory. All these issues manifest primally in CastCIC, the cast calculus used to define GCIC. In this work, we present an extension of CastCIC called GRIP. GRIP is a reasonably gradual type theory that addresses the issues above, featuring internal precision and general exception handling. By adopting a novel interpretation of the unknown term that carefully accounts for universe levels, GRIP satisfies graduality for a large and well-defined class of terms, in addition to being normalizing and a conservative extension of CIC. Internal precision supports reasoning about graduality within GRIP itself, for instance to characterize gradual exception-handling terms, and supports gradual subset types. We develop the metatheory of GRIP using a model formalized in Coq, and provide a prototype implementation of GRIP in Agda.

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Cited By

Malewski MMaillard KTabareau NTanter É(2024)Gradual Indexed Inductive TypesProceedings of the ACM on Programming Languages10.1145/36746448:ICFP(544-572)Online publication date: 15-Aug-2024
https://doi.org/10.1145/3674644
Yuan YGuest SGriffis EPotter HMoon DOmar C(2023)Live Pattern Matching with Typed HolesProceedings of the ACM on Programming Languages10.1145/35860487:OOPSLA1(609-635)Online publication date: 6-Apr-2023
https://dl.acm.org/doi/10.1145/3586048
Eremondi JGarcia RTanter É(2022)Propositional equality for gradual dependently typed programmingProceedings of the ACM on Programming Languages10.1145/35476276:ICFP(165-193)Online publication date: 31-Aug-2022
https://dl.acm.org/doi/10.1145/3547627

Index Terms

A reasonably gradual type theory
1. Theory of computation
  1. Logic
    1. Type theory
  2. Semantics and reasoning
    1. Program constructs
      1. Type structures
    2. Program reasoning

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cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 6, Issue ICFP

August 2022

959 pages

EISSN:2475-1421

DOI:10.1145/3554306

Editor:
Philip Wadler
University of Edinburgh, UK

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Copyright © 2022 Owner/Author.

This work is licensed under a Creative Commons Attribution 4.0 International License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 August 2022

Published in PACMPL Volume 6, Issue ICFP

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Cited By

Malewski MMaillard KTabareau NTanter É(2024)Gradual Indexed Inductive TypesProceedings of the ACM on Programming Languages10.1145/36746448:ICFP(544-572)Online publication date: 15-Aug-2024
https://doi.org/10.1145/3674644
Yuan YGuest SGriffis EPotter HMoon DOmar C(2023)Live Pattern Matching with Typed HolesProceedings of the ACM on Programming Languages10.1145/35860487:OOPSLA1(609-635)Online publication date: 6-Apr-2023
https://dl.acm.org/doi/10.1145/3586048
Eremondi JGarcia RTanter É(2022)Propositional equality for gradual dependently typed programmingProceedings of the ACM on Programming Languages10.1145/35476276:ICFP(165-193)Online publication date: 31-Aug-2022
https://dl.acm.org/doi/10.1145/3547627

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