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Hereditarily Finite Sets in Constructive Type Theory

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Interactive Theorem Proving (ITP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9807))

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Abstract

We axiomatize hereditarily finite sets in constructive type theory and show that all models of the axiomatization are isomorphic. The axiomatization takes the empty set and adjunction as primitives and comes with a strong induction principle. Based on the axiomatization, we construct the set operations of ZF and develop the basic theory of finite ordinals and cardinality. We construct a model of the axiomatization as a quotient of an inductive type of binary trees. The development is carried out in Coq.

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Notes

  1. 1.

    An impredicative definition of R looks as follows: \(\lambda (XY:\mathrm {HF})(x:X)(y:Y). \forall S:X\rightarrow Y\rightarrow \mathsf {Prop}.~~ S\emptyset \emptyset \rightarrow (\forall axby.~Sab\rightarrow Sxy\rightarrow S(a{.}x)(b{.}y))\rightarrow Sxy \).

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Acknowledgement

Denis Müller contributed to the study of tree equivalence during his Bachelor’s thesis project on finitary sets.

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Authors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    Gert Smolka & Kathrin Stark

Authors
  1. Gert Smolka
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  2. Kathrin Stark
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Correspondence to Gert Smolka or Kathrin Stark .

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Editors and Affiliations

  1. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Stephan Merz

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Smolka, G., Stark, K. (2016). Hereditarily Finite Sets in Constructive Type Theory. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_23

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  • DOI: https://doi.org/10.1007/978-3-319-43144-4_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43143-7

  • Online ISBN: 978-3-319-43144-4

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