Journal of Formalized Reasoning
https://jfr.unibo.it/
<strong>Journal of Formalized Reasoning (JFR) – ISSN 1972-5787</strong> encourages submission of papers describing significant, automated or semi-automated formalization efforts in any area, including classical mathematics, constructive mathematics, formal algorithms, and program verification. The emphasis of the journal is on proof techniques and methodologies and their impact on the formalization process. In particular, the journal provides a forum for comparing alternative approaches, enhancing reusability of solutions and offering a clear view of the current state of the field.Alma Mater Studiorum - University of Bolognaen-USJournal of Formalized Reasoning1972-5787<p>Copyrights and publishing rights of all the texts on this journal belong to the respective authors without restrictions.</p><div><a href="http://creativecommons.org/licenses/by/3.0/" rel="license"><img src="https://i.creativecommons.org/l/by/3.0/88x31.png" alt="Creative Commons License" /></a></div><p>This journal is licensed under a <a href="http://creativecommons.org/licenses/by/3.0/" rel="license">Creative Commons Attribution 3.0 Unported License</a> (<a href="http://creativecommons.org/licenses/by/3.0/legalcode">full legal code</a>). <br />See also our <a href="/about/editorialPolicies#openAccessPolicy">Open Access policy</a></p>Implementation of Bourbaki's Elements of Mathematics in Coq: Part Two, From Natural Numbers to Real Numbers
https://jfr.unibo.it/article/view/4771
This paper describes a formalization of the first book of the series ``Elements of Mathematics'' by Nicolas Bourbaki, using the Coq proof assistant. In a first paper published in this journal, we presented the axioms and basic constructions (corresponding to a part of the first two chapters of book I, theory of sets). We discuss here the set of integers (third chapter of book I, theory of set), the sets Z and Q (first chapter of book II, Algebra) and the set of real numbers (Chapter 4 of book III, General topology). We start with a comparison of the Bourbaki approach, the Coq standard library, and the Ssreflect library, then present our implementation.José Grimm
Copyright (c) 2016 José Grimm
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2016-12-012016-12-019215210.6092/issn.1972-5787/4771Formalization of the pumping lemma for context-free languages
https://jfr.unibo.it/article/view/5595
<p>Context-free languages are highly important in computer language processing technology as well as in formal language theory. The Pumping Lemma is a property that is valid for all context-free languages, and is used to show the existence of non context-free languages. This paper presents a formalization, using the Coq proof assistant, of the Pumping Lemma for context-free languages.</p>Marcus V M RamosJosé Carlos Bacelar AlmeidaNelma MoreiraRuy José Guerra Barretto de Queiroz
Copyright (c) 2016 Marcus V M Ramos
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2016-12-012016-12-0192536810.6092/issn.1972-5787/5595Understanding and maintaining tactics graphically OR how we are learning that a diagram can be worth more than 10K LoC
https://jfr.unibo.it/article/view/6298
<p class="p1">The use of a functional language to implement proof strategies as proof tactics in interactive theorem provers, often provides short, concise and elegant implementations. Whilst being elegant, the use of higher order features and combinator languages often results in a very procedural view of a strategy, which may deviate significantly from the high-level ideas behind it. This can make a tactic hard to understand and hence difficult to to debug and maintain for experts and non-experts alike: one often has to tear apart complex combinations of lower level tactics manually in order to analyse a failure in the overall strategy.</p><p class="p1">In an industrial technology transfer project, we have been working on porting a very large and complex proof tactic into PSGraph, a graphical language for representing proof strategies. The goal of this work is to improve understandability and maintainability of tactics. Motivated by some initial successes with this, we here extend PSGraph with additional features for development and debugging. Through the re-implementation and refactoring of several existing tactics, we demonstrates the advantages of PSGraph compared with a typical sentential tactic language with respect to debugging, readability and maintenance. In order to act as guidance for others, we give a fairly detailed comparison of the user experience with the two approaches. The paper is supported by a web page providing further details about the implementation as well as interactive illustrations of the examples.</p>YuHui LinGudmund GrovRob Arthan
Copyright (c) 2016 YuHui Lin, Gudmund Grov, Rob Arthan
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2016-12-152016-12-15926913010.6092/issn.1972-5787/6298Lawvere-Tierney sheafification in Homotopy Type Theory
https://jfr.unibo.it/article/view/6232
<p class="p1">Sheafification is a popular tool in topos theory which allows to extend the internal logic of a topos with new principles. One of its most famous applications is the possibility to transform a topos into a boolean topos using the dense topology, which corresponds in essence to Gödel’s double negation translation. The same construction has not been developed in Martin-Löf type theory because of a mismatch between topos theory and type theory. This mismatched has been fixed recently by considering homotopy type theory, an extension of Martin-Löf type theory with new principles inspired by category theory and homotopy theory, and which corresponds closely to higher toposes. In this paper, we give a computer-checked construction of Lawvere-Tierney sheafification in homotopy type theory.</p>Kevin QuirinNicolas Tabareau
Copyright (c) 2016 Kevin Quirin, Nicolas Tabareau
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2016-12-302016-12-309213116110.6092/issn.1972-5787/6232