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Typing Total Recursive Functions in Coq

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Interactive Theorem Proving (ITP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10499))

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Abstract

We present a (relatively) short mechanized proof that Coq types any recursive function which is provably total in Coq. The well-founded (and terminating) induction scheme, which is the foundation of Coq recursion, is maximal. We implement an unbounded minimization scheme for decidable predicates. It can also be used to reify a whole category of undecidable predicates. This development is purely constructive and requires no axiom. Hence it can be integrated into any project that might assume additional axioms.

Work partially supported by the TICAMORE project (ANR grant 16-CE91-0002).

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Notes

  1. 1.

    From which the term \(t_f \,{\mathtt {:=}}\,\varvec{v}\,{\mapsto }\,{\mathtt {proj1\_sig}}(Q\;\varvec{v})\) of (CiT) is trivially derived.

  2. 2.

    It is difficult to use a word more accurate than “seems” because the relevant discussion in [2] is just a short outline of an approach, not a proof or an actual implementation.

  3. 3.

    Of course this statement is of philosophical nature. We do not claim that assuming additional axiom is evil, but carelessly adding axioms is a recipe for inconsistencies.

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Authors and Affiliations

  1. LORIA – CNRS, Nancy, France

    Dominique Larchey-Wendling

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  1. Dominique Larchey-Wendling
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Correspondence to Dominique Larchey-Wendling .

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Editors and Affiliations

  1. University of Brasília, Brasília D.F., Brazil

    Mauricio Ayala-Rincón

  2. NASA, Hampton, VA, USA

    César A. Muñoz

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Larchey-Wendling, D. (2017). Typing Total Recursive Functions in Coq. In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_24

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  • DOI: https://doi.org/10.1007/978-3-319-66107-0_24

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66106-3

  • Online ISBN: 978-3-319-66107-0

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