Automatically proving equivalence by type-safe reflection

Franck Slama, Edwin Charles Brady

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)
1 Downloads (Pure)

Abstract

One difficulty with reasoning and programming with dependent types is that proof obligations arise naturally once programs become even moderately sized. For example, implementing an adder for binary numbers indexed over their natural number equivalents naturally leads to proof obligations for equalities of expressions over natural numbers. The need for these equality proofs comes, in intensional type theories, from the fact that the propositional equality enables us to prove as equal terms that are not judgementally equal, which means that the typechecker can’t always obtain equalities by reduction. As far as possible, we would like to solve such proof obligations automatically. In this paper, we show one way to automate these proofs by reflection in the dependently typed programming language Idris. We show how defining reflected terms indexed by the original Idris expression allows us to construct and manipulate proofs. We build a hierarchy of tactics for proving equivalences in semi-groups, monoids, commutative monoids, groups, commutative groups, semi-rings and rings. We also show how each tactic reuses those from simpler structures, thus avoiding duplication of code and proofs.
Original languageEnglish
Title of host publicationIntelligent Computer Mathematics
Subtitle of host publication10th International Conference, CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceedings
EditorsHerman Geuvers, Matthew England, Osman Hasan, Florian Rabe, Olaf Teschke
Place of PublicationCham
PublisherSpringer
Pages40-55
Number of pages16
ISBN (Electronic)9783319620756
ISBN (Print)9783319620749
DOIs
Publication statusPublished - 2017
Event10th Conference on Intelligent Computer Mathematics (CICM 2017) - Edinburgh, United Kingdom
Duration: 17 Jul 201721 Jul 2017
Conference number: 10
http://www.cicm-conference.org/2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence)
PublisherSpringer
Volume10383 (LNCS)
ISSN (Print)0302-9743

Conference

Conference10th Conference on Intelligent Computer Mathematics (CICM 2017)
Abbreviated titleCICM
Country/TerritoryUnited Kingdom
CityEdinburgh
Period17/07/1721/07/17
Internet address

Keywords

  • Proof automation
  • Equivalence
  • Equality
  • Proof by reflection
  • Correct-by-construction software
  • Type-driven development

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