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Theorem Proving Modulo

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Abstract

Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of general interest because it permits one to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof-theoretic account of the combination of computations and deductions. The congruence on propositions is handled through rewrite rules and equational axioms. Rewrite rules apply to terms but also directly to atomic propositions.

The second contribution is to give a complete proof search method, called extended narrowing and resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo.

An important application is that higher-order logic can be presented as a theory in deduction modulo. Applying the ENAR method to this presentation of higher-order logic subsumes full higher-order resolution.

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Authors and Affiliations

  1. Ecole polytechnique and INRIA LIX, Ecole polytechnique, 91128, Palaiseau Cedex, France

    Gilles Dowek

  2. LIP6 and INRIA-Rocquencourt, Université Paris 6, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Thérèse Hardin

  3. LORIA and INRIA, 615, rue du Jardin Botanique, 54600, Villers-lès-Nancy Cedex, France

    Claude Kirchner

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  1. Gilles Dowek
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  2. Thérèse Hardin
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  3. Claude Kirchner
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Dowek, G., Hardin, T. & Kirchner, C. Theorem Proving Modulo. Journal of Automated Reasoning 31, 33–72 (2003). https://doi.org/10.1023/A:1027357912519

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