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Congruence Closure in Intensional Type Theory

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Automated Reasoning (IJCAR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9706))

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Abstract

Congruence closure procedures are used extensively in automated reasoning and are a core component of most satisfiability modulo theories solvers. However, no known congruence closure algorithms can support any of the expressive logics based on intensional type theory (ITT), which form the basis of many interactive theorem provers. The main source of expressiveness in these logics is dependent types, and yet existing congruence closure procedures found in interactive theorem provers based on ITT do not handle dependent types at all and only work on the simply-typed subsets of the logics. Here we present an efficient and proof-producing congruence closure procedure that applies to every function in ITT no matter how many dependencies exist among its arguments, and that only relies on the commonly assumed uniqueness of identity proofs axiom. We demonstrate its usefulness by solving interesting verification problems involving functions with dependent types.

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Notes

  1. 1.

    There are many equivalent ways of defining heq. One popular way is “John Major equality” [15]. Additional formulations and formal proofs of equivalence can be found at http://leanprover.github.io/ijcar16/congr.lean.

  2. 2.

    The formal statements and proofs for small values of n can be found at http://leanprover.github.io/ijcar16/congr.lean, along with formal proofs of all other lemmas described in this paper.

  3. 3.

    To simplify the presentation further, we ignore the possibility that any of these subterms themselves include partial applications of equality.

  4. 4.

    https://github.com/leanprover/lean/blob/master/src/library/blast/congruence_closure.cpp.

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Acknowledgments

We would like to thank David Dill, Jeremy Avigad, Robert Lewis, Nikhil Swany, Floris van Doorn and Georges Gonthier for providing valuable feedback on early drafts.

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Authors and Affiliations

  1. Stanford University, Stanford, USA

    Daniel Selsam

  2. Microsoft Research, Redmond, USA

    Leonardo de Moura

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  1. Daniel Selsam
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Correspondence to Leonardo de Moura .

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  1. Aix-Marseille University, Marseille, France

    Nicola Olivetti

  2. SRI International, Menlo Park, California, USA

    Ashish Tiwari

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Selsam, D., de Moura, L. (2016). Congruence Closure in Intensional Type Theory. In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_8

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  • DOI: https://doi.org/10.1007/978-3-319-40229-1_8

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