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Type Theories from Barendregt’s Cube for Theorem Provers

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Advances in Natural Deduction

Part of the book series: Trends in Logic ((TREN,volume 39))

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Abstract

Anybody using a theorem prover or proof assistant will want to have confidence that the system involved will not permit the derivation of false results. On some occasions, there is more than the usual need for this confidence. This chapter discusses some logical systems based on typed lambda-calculus that can be used for this purpose. The systems are natural deduction systems, and use the propositions-as-types paradigm. Not only are the underlying systems provably consistent, but additional unproved assumptions from which a lot of ordinary mathematics can be derived can also be proved consistent. Finally, the systems have few primitive postulates that need to be programmed separately, so that it is easier for a programmer to see whether the code really does program the systems involved without errors.

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Notes

  1. 1.

    It is common to use * for \({\mathsf {Prop}}\), \(\Box \) for \({\mathsf {Type}}\). I formerly referred to sorts as kinds [21–25].

  2. 2.

    Compare this to the system Nuprl [10], which has over one hundred primitive postulates, each of which must be programmed separately in implementation.

  3. 3.

    See [22, Sect. 6].

  4. 4.

    It can be shown that every type in a well-formed environment converts a term of this form.

  5. 5.

    The rule for \(\eta \)-reduction is that \({\lambda }x : A {\;.\;}Ux \rhd U\) if \(x\) is not free in \(U\).

  6. 6.

    As I write this, I have not yet found a general way of writing this predicate, but I have succeeded for an example which is not covariant.

  7. 7.

    The adaptation of this proof seems to work in the case of the example mentioned in the preceding footnote.

  8. 8.

    The definition will also work if \(U : {{\mathsf {Type}}}\), but this is not needed here.

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Acknowledgments

This work was supported in part by grant RGP-23391-98 from the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada

    Jonathan P. Seldin

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  1. Jonathan P. Seldin
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Correspondence to Jonathan P. Seldin .

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  1. Departamento de Filosofia, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil

    Luiz Carlos Pereira

  2. Departamento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil

    Edward Hermann Haeusler

  3. School of Computer Science, University of Birmingham, Birmingham, United Kingdom

    Valeria de Paiva

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Seldin, J.P. (2014). Type Theories from Barendregt’s Cube for Theorem Provers. In: Pereira, L., Haeusler, E., de Paiva, V. (eds) Advances in Natural Deduction. Trends in Logic, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7548-0_7

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