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Automatically Proving Equivalence by Type-Safe Reflection

  • Franck Slama
  • Edwin Brady
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)

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.

Keywords

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

Notes

Acknowledgements

We thank the anonymous reviewers and Jacques Carette for their insightful comments on an earlier draft. We are also grateful for the support of the Scottish Informatics and Computer Science Alliance (SICSA) and EPSRC grant EP/N024222/1.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of St AndrewsSt AndrewsScotland, UK

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