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SMLtoCoq: Automated Generation of Coq Specifications and Proof Obligations from SML Programs with Contracts 从带有契约的SML程序中自动生成Coq规范和证明义务
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.6
Laila El-Beheiry, Giselle Reis, Ammar Karkour
{"title":"SMLtoCoq: Automated Generation of Coq Specifications and Proof Obligations from SML Programs with Contracts","authors":"Laila El-Beheiry, Giselle Reis, Ammar Karkour","doi":"10.4204/EPTCS.337.6","DOIUrl":"https://doi.org/10.4204/EPTCS.337.6","url":null,"abstract":"Formally reasoning about functional programs is supposed to be straightforward and elegant, however, it is not typically done as a matter of course. Reasoning in a proof assistant requires\"reimplementing\"the code in those tools, which is far from trivial. SMLtoCoq provides an automatic translation of SML programs and function contracts into Coq. Programs are translated into Coq specifications, and function contracts into theorems, which can then be formally proved. Using the Equations plugin and other well established Coq libraries, SMLtoCoq is able to translate SML programs without side-effects containing partial functions, structures, functors, records, among others. Additionally, we provide a Coq version of many parts of SML's basis library, so that calls to these libraries are kept almost as is.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121486633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Touring the MetaCoq Project (Invited Paper) 参观MetaCoq项目(特邀论文)
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.2
Matthieu Sozeau
{"title":"Touring the MetaCoq Project (Invited Paper)","authors":"Matthieu Sozeau","doi":"10.4204/EPTCS.337.2","DOIUrl":"https://doi.org/10.4204/EPTCS.337.2","url":null,"abstract":"Proof assistants are getting more widespread use in research and industry to provide certified and independently checkable guarantees about theories, designs, systems and implementations. However, proof assistant implementations themselves are seldom verified, although they take a major share of the trusted code base in any such certification effort. In this area, proof assistants based on Higher-Order Logic enjoy stronger guarantees, as self-certified implementations have been available for some years. One cause of this difference is the inherent complexity of dependent type theories together with their extensions with inductive types, universe polymorphism and complex sort systems, and the gap between theory on paper and practical implementations in efficient programming languages. MetaCoq is a collaborative project that aims to tackle these difficulties to provide the first fully-certified realistic implementation of a type checker for the full calculus underlying the Coq proof assistant. To achieve this, we refined the sometimes blurry, if not incorrect, specification and implementation of the system. We show how theoretical tools from this community such as bidirectional type-checking, Tait-Martin-L\"of/Takahashi's confluence proof technique and monadic and dependently-typed programming can help construct the following artefacts: a specification of Coq's syntax and type theory, the Polymorphic Cumulative Calculus of (Co)-Inductive Constructions (PCUIC); a monad for the manipulation of raw syntax and interaction with the Coq system; a verification of PCUIC's metatheory, whose main results are the confluence of reduction, type preservation and principality of typing; a realistic, correct and complete type-checker for PCUIC; a sound type and proof erasure procedure from PCUIC to untyped lambda-calculus, i.e., the core of the extraction mechanism of Coq.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"279 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116273370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Interacting Safely with an Unsafe Environment 与不安全的环境进行安全交互
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.3
Gilles Dowek
{"title":"Interacting Safely with an Unsafe Environment","authors":"Gilles Dowek","doi":"10.4204/EPTCS.337.3","DOIUrl":"https://doi.org/10.4204/EPTCS.337.3","url":null,"abstract":"We give a presentation of Pure type systems where contexts need not be well-formed and show that this presentation is equivalent to the usual one. The main motivation for this presentation is that, when we extend Pure type systems with computation rules, like in the logical framework Dedukti, we want to declare the constants before the computation rules that are needed to check the well-typedness of their type.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130389437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Facilitating Meta-Theory Reasoning (Invited Paper) 促进元理论推理(特邀论文)
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.1
Giselle Reis
{"title":"Facilitating Meta-Theory Reasoning (Invited Paper)","authors":"Giselle Reis","doi":"10.4204/EPTCS.337.1","DOIUrl":"https://doi.org/10.4204/EPTCS.337.1","url":null,"abstract":"Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be designed as a set of rules such that proofs using those rules are correct by construction. Therefore, one must consider all ways these rules can interact and prove that they satisfy certain properties which makes them\"well-behaved\". This is called the meta-theory of a proof system. Meta-theory proofs typically involve many cases on structures with lots of symbols. The majority of cases are usually quite similar, and when a proof fails, it might be because of a sub-case on a very specific configuration of rules. Developing these proofs by hand is tedious and error-prone, and their combinatorial nature suggests they could be automated. There are various approaches on how to automate, either partially or completely, meta-theory proofs. In this paper, I will present some techniques that I have been involved in for facilitating meta-theory reasoning.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126268517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Countability of Inductive Types Formalized in the Object-Logic Level 在对象逻辑层形式化的归纳类型的可数性
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.5
Qinxiang Cao, Xiwei Wu
{"title":"Countability of Inductive Types Formalized in the Object-Logic Level","authors":"Qinxiang Cao, Xiwei Wu","doi":"10.4204/EPTCS.337.5","DOIUrl":"https://doi.org/10.4204/EPTCS.337.5","url":null,"abstract":"The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many elements. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this proof, we provide a Coq tactic to automatically prove the countability of concrete inductive types. We also developed Coq libraries for countability and for the syntax of inductive definitions, which have value on their own.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127293740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Systematic Translation of Formalizations of Type Theory from Intrinsic to Extrinsic Style 类型理论形式化从内在风格到外在风格的系统翻译
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-07-16 DOI: 10.4204/EPTCS.337.7
Florian Rabe, Navid Roux
{"title":"Systematic Translation of Formalizations of Type Theory from Intrinsic to Extrinsic Style","authors":"Florian Rabe, Navid Roux","doi":"10.4204/EPTCS.337.7","DOIUrl":"https://doi.org/10.4204/EPTCS.337.7","url":null,"abstract":"Type theories can be formalized using the intrinsically (hard) or the extrinsically (soft) typed style. In large libraries of type theoretical features, often both styles are present, which can lead to code duplication and integration issues. We define an operator that systematically translates a hard-typed into the corresponding soft-typed formulation. Even though this translation is known in principle, a number of subtleties make it more difficult than naively expected. Importantly, our translation preserves modularity, i.e., it maps structured sets of hard-typed features to correspondingly structured soft-typed ones. We implement our operator in the MMT system and apply it to a library of type-theoretical features.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114408925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading 具有Ad-hoc过载的HOL模型理论保守扩展的机械化
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-01-11 DOI: 10.4204/EPTCS.332.1
A. Gengelbach, Johannes Åman Pohjola, Tjark Weber
{"title":"Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading","authors":"A. Gengelbach, Johannes Åman Pohjola, Tjark Weber","doi":"10.4204/EPTCS.332.1","DOIUrl":"https://doi.org/10.4204/EPTCS.332.1","url":null,"abstract":"Definitions of new symbols merely abbreviate expressions in logical frameworks, and no new facts (regarding previously defined symbols) should hold because of a new definition. In Isabelle/HOL, definable symbols are types and constants. The latter may be ad-hoc overloaded, i.e. have different definitions for non-overlapping types. We prove that symbols that are independent of a new definition may keep their interpretation in a model extension. This work revises our earlier notion of model-theoretic conservative extension and generalises an earlier model construction. We obtain consistency of theories of definitions in higher-order logic (HOL) with ad-hoc overloading as a corollary. Our results are mechanised in the HOL4 theorem prover.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131109452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory 两层类型理论在Dedukti中的实现及其在立方类型理论中的应用
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-01-11 DOI: 10.4204/EPTCS.332.4
Bruno Barras, Valentin Maestracci
{"title":"Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory","authors":"Bruno Barras, Valentin Maestracci","doi":"10.4204/EPTCS.332.4","DOIUrl":"https://doi.org/10.4204/EPTCS.332.4","url":null,"abstract":"In this paper, we make a substantial step towards an encoding of Cubical Type Theory (CTT) in the Dedukti logical framework. Type-checking CTT expressions features a decision procedure in a de Morgan algebra that so far could not be expressed by the rewrite rules of Dedukti. As an alternative, 2 Layer Type Theories are variants of Martin-L\"of Type Theory where all or part of the definitional equality can be represented in terms of a so-called external equality. We propose to split the encoding by giving an encoding of 2 Layer Type Theories (2LTT) in Dedukti, and a partial encoding of CTT in 2LTT.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132980320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Object-Level Reasoning with Logics Encoded in HOL Light 用HOL Light编码的逻辑进行对象级推理
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2021-01-11 DOI: 10.4204/EPTCS.332.2
P. Papapanagiotou, Jacques D. Fleuriot
{"title":"Object-Level Reasoning with Logics Encoded in HOL Light","authors":"P. Papapanagiotou, Jacques D. Fleuriot","doi":"10.4204/EPTCS.332.2","DOIUrl":"https://doi.org/10.4204/EPTCS.332.2","url":null,"abstract":"We present a generic framework that facilitates object level reasoning with logics that are encoded within the Higher Order Logic theorem proving environment of HOL Light. This involves proving statements in any logic using intuitive forward and backward chaining in a sequent calculus style. It is made possible by automated machinery that take care of the necessary structural reasoning and term matching automatically. Our framework can also handle type theoretic correspondences of proofs, effectively allowing the type checking and construction of computational processes via proof. We demonstrate our implementation using a simple propositional logic and its Curry-Howard correspondence to the lambda-calculus, and argue its use with linear logic and its various correspondences to session types.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128345774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Deductive Systems and Coherence for Skew Prounital Closed Categories 斜边闭范畴的演绎系统与相干性
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2020-06-01 DOI: 10.4204/EPTCS.332.3
Tarmo Uustalu, Niccolò Veltri, N. Zeilberger
{"title":"Deductive Systems and Coherence for Skew Prounital Closed Categories","authors":"Tarmo Uustalu, Niccolò Veltri, N. Zeilberger","doi":"10.4204/EPTCS.332.3","DOIUrl":"https://doi.org/10.4204/EPTCS.332.3","url":null,"abstract":"In this paper we develop the proof theory of skew prounital closed categories. These are variants of the skew closed categories of Street where the unit is not represented. Skew closed categories in turn are a weakening of the closed categories of Eilenberg and Kelly where no structural law is required to be invertible. The presence of a monoidal structure in these categories is not required. We construct several equivalent presentations of the free skew prounital closed category on a given set of generating objects: a categorical calculus (Hilbert-style system), a cut-free sequent calculus and a natural deduction system corresponding to a variant of planar (= non-commutative linear) typed lambda-calculus. We solve the coherence problem for skew prounital closed categories by showing that the sequent calculus admits focusing and presenting two reduction-free normalization procedures for the natural deduction calculus: normalization by evaluation and hereditary substitutions. Normal natural deduction derivations (βη-long forms) are in one-to-one correspondence with derivations in the focused sequent calculus. Unexpectedly, the free skew prounital closed category on a set satisfies a left-normality condition which makes it lose its skew aspect. This pitfall can be avoided by considering the free skew prounital closed category on a skew multicategory instead. The latter has a presentation as a cut-free sequent calculus for which it is easy to see that the left-normality condition generally fails. The whole development has been fully formalized in the dependently typed programming language Agda.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132180537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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