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DOI: 10.4230/LIPIcs.TYPES.2022.11

Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families

Authors: Catherine Dubois, Nicolas Magaud, and Alain Giorgetti

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract

There are several ways to formally represent families of data, such as lambda terms, in a type theory such as the dependent type theory of Coq. Mathematical representations are very compact ones and usually rely on the use of dependent types, but they tend to be difficult to handle in practice. On the contrary, implementations based on a larger (and simpler) data structure combined with a restriction property are much easier to deal with. In this work, we study several families related to lambda terms, among which Motzkin trees, seen as lambda term skeletons, closable Motzkin trees, corresponding to closed lambda terms, and a parameterized family of open lambda terms. For each of these families, we define two different representations, show that they are isomorphic and provide tools to switch from one representation to another. All these datatypes and their associated transformations are implemented in the Coq proof assistant. Furthermore we implement random generators for each representation, using the QuickChick plugin.

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Catherine Dubois, Nicolas Magaud, and Alain Giorgetti. Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dubois_et_al:LIPIcs.TYPES.2022.11,
  author =	{Dubois, Catherine and Magaud, Nicolas and Giorgetti, Alain},
  title =	{{Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.11},
  URN =		{urn:nbn:de:0030-drops-184548},
  doi =		{10.4230/LIPIcs.TYPES.2022.11},
  annote =	{Keywords: Data Representations, Isomorphisms, dependent Types, formal Proofs, random Generation, lambda Terms, Coq}
}

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DOI: 10.4230/LIPIcs.RTA.2013.303

Automatic Decidability: A Schematic Calculus for Theories with Counting Operators

Authors: Elena Tushkanova, Christophe Ringeissen, Alain Giorgetti, and Olga Kouchnarenko

Published in: LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)

Abstract

Many verification problems can be reduced to a satisfiability problem modulo theories. For building satisfiability procedures the rewriting-based approach uses a general calculus for equational reasoning named paramodulation. Schematic paramodulation, in turn, provides means to reason on the derivations computed by paramodulation. Until now, schematic paramodulation was only studied for standard paramodulation. We present a schematic paramodulation calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories equipped with counting operators. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records. An implementation within the rewriting-based Maude system constitutes a practical contribution. It enables automatic decidability proofs for theories of practical use.

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Elena Tushkanova, Christophe Ringeissen, Alain Giorgetti, and Olga Kouchnarenko. Automatic Decidability: A Schematic Calculus for Theories with Counting Operators. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 303-318, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{tushkanova_et_al:LIPIcs.RTA.2013.303,
  author =	{Tushkanova, Elena and Ringeissen, Christophe and Giorgetti, Alain and Kouchnarenko, Olga},
  title =	{{Automatic Decidability: A Schematic Calculus for Theories with Counting Operators}},
  booktitle =	{24th International Conference on Rewriting Techniques and Applications (RTA 2013)},
  pages =	{303--318},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-53-8},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{21},
  editor =	{van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.303},
  URN =		{urn:nbn:de:0030-drops-40696},
  doi =		{10.4230/LIPIcs.RTA.2013.303},
  annote =	{Keywords: decision procedures, superposition, schematic saturation}
}

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Documents authored by Giorgetti, Alain

Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families

Abstract

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Automatic Decidability: A Schematic Calculus for Theories with Counting Operators

Abstract

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