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DOI: 10.4230/LIPIcs.ITP.2019.14

First-Order Guarded Coinduction in Coq

Authors: Łukasz Czajka

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract

We introduce two coinduction principles and two proof translations which, under certain conditions, map coinductive proofs that use our principles to guarded Coq proofs. The first principle provides an "operational" description of a proof by coinduction, which is easy to reason with informally. The second principle extends the first one to allow for direct proofs by coinduction of statements with existential quantifiers and multiple coinductive predicates in the conclusion. The principles automatically enforce the correct use of the coinductive hypothesis. We implemented the principles and the proof translations in a Coq plugin.

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Łukasz Czajka. First-Order Guarded Coinduction in Coq. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{czajka:LIPIcs.ITP.2019.14,
  author =	{Czajka, {\L}ukasz},
  title =	{{First-Order Guarded Coinduction in Coq}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.14},
  URN =		{urn:nbn:de:0030-drops-110690},
  doi =		{10.4230/LIPIcs.ITP.2019.14},
  annote =	{Keywords: coinduction, Coq, guardedness, corecursion}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.12

Polymorphic Higher-Order Termination

Authors: Łukasz Czajka and Cynthia Kop

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract

We generalise the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism. Instead of using weakly monotonic functionals, we interpret terms in a suitable extension of System F_omega. This enables a direct interpretation of rewrite rules which make essential use of impredicative polymorphism. In addition, our generalisation eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types. As an illustration of the potential of our method, we prove termination of a substantial fragment of full intuitionistic second-order propositional logic with permutative conversions.

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Łukasz Czajka and Cynthia Kop. Polymorphic Higher-Order Termination. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{czajka_et_al:LIPIcs.FSCD.2019.12,
  author =	{Czajka, {\L}ukasz and Kop, Cynthia},
  title =	{{Polymorphic Higher-Order Termination}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.12},
  URN =		{urn:nbn:de:0030-drops-105193},
  doi =		{10.4230/LIPIcs.FSCD.2019.12},
  annote =	{Keywords: termination, polymorphism, higher-order rewriting, permutative conversions}
}

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Documents authored by Czajka, Łukasz

First-Order Guarded Coinduction in Coq

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Polymorphic Higher-Order Termination

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