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Superposition with Structural Induction

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Frontiers of Combining Systems (FroCoS 2017)

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Abstract

Superposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved. We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging \(\text {AVATAR}\) [1]. Lemmas obtained by generalization or theory exploration can be introduced during search, used, and proved, all in the same search space. We describe an implementation and present some promising results.

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Notes

  1. 1.

    It might even induce rewriting loops in some cases where the term ordering used by superposition and the rewrite system are not compatible. In our experience this does not seem to happen often.

  2. 2.

    Our framework allows attempting to prove several distinct inductive goals to solve a single subgoal.

  3. 3.

    CVC4 1.5-prerelease r6317, see http://cvc4.cs.stanford.edu/web/ .

  4. 4.

    Commit 187b71af8d920d0634b2b8b34c4ac4834b2f6a94 at https://github.com/tip-org/benchmarks.

  5. 5.

    Experiments on TPTP were run on a 2.20 GHz Intel \(\text {Xeon}^\circledR \) CPU with 30 s timeout and a memory limit of 2 GB.

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Acknowledgments

The author would like to thank Jasmin Blanchette, Gilles Dowek, Guillaume Burel, Pascal Fontaine, and reviewers of previous versions of this paper (one of them, in particular, for pointing out a lot of related works and limitations in several occasions).

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  1. University of Lorraine, CNRS, Inria, LORIA, 54000, Nancy, France

    Simon Cruanes

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  1. Simon Cruanes
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Correspondence to Simon Cruanes .

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  1. University of Liverpool, Liverpool, United Kingdom

    Clare Dixon

  2. University of Sao Paulo, Sao Paulo, Brazil

    Marcelo Finger

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Cruanes, S. (2017). Superposition with Structural Induction. In: Dixon, C., Finger, M. (eds) Frontiers of Combining Systems. FroCoS 2017. Lecture Notes in Computer Science(), vol 10483. Springer, Cham. https://doi.org/10.1007/978-3-319-66167-4_10

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  • DOI: https://doi.org/10.1007/978-3-319-66167-4_10

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