79 research outputs found

    A simplified framework for first-order languages and its formalization in Mizar

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    A strictly formal, set-theoretical treatment of classical first-order logic is given. Since this is done with the goal of a concrete Mizar formalization of basic results (Lindenbaum lemma; Henkin, satisfiability, completeness and Lowenheim-Skolem theorems) in mind, it turns into a systematic pursue of simplification: we give up the notions of free occurrence, of derivation tree, and study what inference rules are strictly needed to prove the mentioned results. Afterwards, we discuss details of the actual Mizar implementation, and give general techniques developed therein.Comment: Ph.D. thesis, defended on January, 20th, 201

    Improving legibility of natural deduction proofs is not trivial

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    In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page

    Case Studies in Proof Checking

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    The aim of computer proof checking is not to find proofs, but to verify them. This is different from automated deduction, which is the use of computers to find proofs that humans have not devised first. Currently, checking a proof by computer is done by taking a known mathematical proof and entering it into the special language recognized by a proof verifier program, and then running the verifier to hopefully obtain no errors. Of course, if the proof checker approves the proof, there are considerations of whether or not the proof checker is correct, and this has been complicated by the fact that so many systems have sprung into being. The two main challenges in using a proof checker today are the time needed to learn the syntax and general usage of the system and the time needed to formalize a proof in the system even when the user is already proficient with it. As mathematicians are not yet using proof checkers regularly, we wanted to evaluate the validity of this reluctance by analyzing these main obstacles. Judging by Dr. Wiedijk’s Formalizing 100 Theorems list, which gives an overview of the headway various proof systems have made in mathematics, Coq and Mizar are two of the most successful systems in use today (Wiedijk, 2007). I simultaneously formalized two fairly involved theorems in these two systems while I was at approximately the same level of familiarity with each. I kept track of my experiences with learning the systems and analyzed their comparative strengths and weaknesses. The analysis and summary of experiences should also give a general idea of the current state of computer-aided proof checking

    LISA - A Modern Proof System

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    Calculational Proofs in ACL2s

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    Teaching college students how to write rigorous proofs is a critical objective in courses that introduce formal reasoning. Over the course of several years, we have developed a mechanically-checkable style of calculational reasoning that we used to teach over a thousand freshman-level undergraduate students how to reason about computation in our "Logic and Computation" class at Northeastern University. We were inspired by Dijkstra, who advocated the use of calculational proofs, writing "calculational proofs are almost always more effective than all informal alternatives, ..., the design of calculational proofs seems much more teachable than the elusive art of discovering an informal proof." Our calculational proof checker is integrated into ACL2s and is available as an Eclipse IDE plugin, via a Web interface, and as a stand-alone tool. It automatically checks proofs for correctness and provides useful feedback. We describe the architecture of the checker, its proof format, its underlying algorithms, its correctness and provide examples using proofs from our undergraduate class and from Dijkstra. We also describe our experiences using the proof checker to teach undergraduates how to formally reason about computation

    TLA+ Proofs

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    TLA+ is a specification language based on standard set theory and temporal logic that has constructs for hierarchical proofs. We describe how to write TLA+ proofs and check them with TLAPS, the TLA+ Proof System. We use Peterson's mutual exclusion algorithm as a simple example to describe the features of TLAPS and show how it and the Toolbox (an IDE for TLA+) help users to manage large, complex proofs.Comment: A shorter version of this article appeared in the proceedings of the conference Formal Methods 2012 (FM 2012, Paris, France, Springer LNCS 7436, pp. 147-154

    Formalization of Real Analysis: A Survey of Proof Assistants and Libraries

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    International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis
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