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Bar Induction is Compatible with Constructive Type Theory

Published: 24 April 2019 Publication History

Abstract

Powerful yet effective induction principles play an important role in computing, being a paramount component of programming languages, automated reasoning, and program verification systems. The Bar Induction (BI) principle is a fundamental concept of intuitionism, which is equivalent to the standard principle of transfinite induction. In this work, we investigate the compatibility of several variants of BI with Constructive Type Theory (CTT), a dependent type theory in the spirit of Martin-Löf’s extensional theory. We first show that CTT is compatible with a BI principle for sequences of numbers. Then, we establish the compatibility of CTT with a more general BI principle for sequences of name-free closed terms. The formalization of the latter principle within the theory involved enriching CTT’s term syntax with a limit constructor and showing that consistency is preserved. Furthermore, we provide novel insights regarding BI, such as the non-truncated version of BI on monotone bars being intuitionistically false. These enhancements are carried out formally using the Nuprl proof assistant that implements CTT and the formalization of CTT within the Coq proof assistant presented in previous works.

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      cover image Journal of the ACM
      Journal of the ACM  Volume 66, Issue 2
      April 2019
      260 pages
      ISSN:0004-5411
      EISSN:1557-735X
      DOI:10.1145/3318168
      Issue’s Table of Contents
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      Publication History

      Published: 24 April 2019
      Accepted: 01 January 2019
      Revised: 01 November 2018
      Received: 01 March 2018
      Published in JACM Volume 66, Issue 2

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      Author Tags

      1. Bar induction
      2. Coq
      3. Nuprl
      4. W types
      5. choice sequences
      6. computational type theory
      7. intuitionistic logic
      8. semantics

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      • Eric and Wendy Schmidt Postdoctoral Award program for Women in Mathematical and Computing Sciences
      • Fonds National de la Recherche Luxembourg
      • Weizmann Institute of Science -- National Postdoctoral Award program for Advancing Women in Science
      • Fulbright Post-doctoral Scholar program

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