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Rewriting Modulo βin the λΠ-Calculus Modulo 在λΠ-Calculus Modulo中重写Modulo β
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2015-07-29 DOI: 10.4204/EPTCS.185.6
Ronan Saillard
{"title":"Rewriting Modulo βin the λΠ-Calculus Modulo","authors":"Ronan Saillard","doi":"10.4204/EPTCS.185.6","DOIUrl":"https://doi.org/10.4204/EPTCS.185.6","url":null,"abstract":"The λ Π-calculus Modulo is a variant of the λ-calculus with dependent types where β-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the λ Π-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with β-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the β-reduction and rewrite rules with λ-abstraction on their left-hand side, we introduce a notion of rewriting modulo β for the λ Π-calculus Modulo. We prove that confluence of rewriting modulo β is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the λ Π-calculus Modulo into Higher-Order Rewrite System (HRS). As a consequence, we also make the confluence results for HRSs available for the λ Π-calculus Modulo.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125780962","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: 4
Gluing together Proof Environments: Canonical extensions of LF Type Theories featuring Locks 粘合证明环境:以锁为特征的LF类型理论的规范扩展
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2015-07-29 DOI: 10.4204/EPTCS.185.1
F. Honsell, L. Liquori, P. Maksimovic, Ivan Scagnetto
{"title":"Gluing together Proof Environments: Canonical extensions of LF Type Theories featuring Locks","authors":"F. Honsell, L. Liquori, P. Maksimovic, Ivan Scagnetto","doi":"10.4204/EPTCS.185.1","DOIUrl":"https://doi.org/10.4204/EPTCS.185.1","url":null,"abstract":"We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can either be invoked in order to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF_P , is the canonical version of the system LLF_P , presented earlier by the authors. The second system, CLLF_P? , features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122563190","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
A Case Study on Logical Relations using Contextual Types 基于上下文类型的逻辑关系案例研究
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2015-07-27 DOI: 10.4204/EPTCS.185.3
Andrew Cave, B. Pientka
{"title":"A Case Study on Logical Relations using Contextual Types","authors":"Andrew Cave, B. Pientka","doi":"10.4204/EPTCS.185.3","DOIUrl":"https://doi.org/10.4204/EPTCS.185.3","url":null,"abstract":"Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131124503","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: 14
An Open Challenge Problem Repository for Systems Supporting Binders 支持绑定的系统的开放挑战问题存储库
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2015-07-27 DOI: 10.4204/EPTCS.185.2
A. Felty, A. Momigliano, B. Pientka
{"title":"An Open Challenge Problem Repository for Systems Supporting Binders","authors":"A. Felty, A. Momigliano, B. Pientka","doi":"10.4204/EPTCS.185.2","DOIUrl":"https://doi.org/10.4204/EPTCS.185.2","url":null,"abstract":"A variety of logical frameworks support the use of higher-order abstract syntax in representing formal systems; however, each system has its own set of benchmarks. Even worse, general proof assistants that provide special libraries for dealing with binders offer a very limited evaluation of such libraries, and the examples given often do not exercise and stress-test key aspects that arise in the presence of binders. In this paper we design an open repository ORBI (Open challenge problem Repository for systems supporting reasoning with BInders). We believe the field of reasoning about languages with binders has matured, and a common set of benchmarks provides an important basis for evaluation and qualitative comparison of different systems and libraries that support binders, and it will help to advance the field.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127627668","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: 7
Sequent Calculus and Equational Programming 序代微积分与方程规划
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2015-07-01 DOI: 10.4204/EPTCS.185.7
Nicolas Guenot, Daniel Gustafsson
{"title":"Sequent Calculus and Equational Programming","authors":"Nicolas Guenot, Daniel Gustafsson","doi":"10.4204/EPTCS.185.7","DOIUrl":"https://doi.org/10.4204/EPTCS.185.7","url":null,"abstract":"Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132240147","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
Automatically Deriving Schematic Theorems for Dynamic Contexts 动态环境下原理图定理的自动推导
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2014-07-17 DOI: 10.1145/2631172.2631181
Olivier Savary Bélanger, Kaustuv Chaudhuri
{"title":"Automatically Deriving Schematic Theorems for Dynamic Contexts","authors":"Olivier Savary Bélanger, Kaustuv Chaudhuri","doi":"10.1145/2631172.2631181","DOIUrl":"https://doi.org/10.1145/2631172.2631181","url":null,"abstract":"Hypothetical judgments go hand-in-hand with higher-order abstract syntax for meta-theoretic reasoning. The dynamic assumptions of these judgments often have a simple regular structure of repetitions of related assumptions; reflecting on this regular structure can let us derive a number of structural properties about the elements of the context automatically. We present an extension of the Abella theorem prover with a new mechanism for defining particular kinds of regular context relations, called schemas, and tacticals to derive theorems from these schemas as needed. Importantly, our extension leaves the trusted kernel of Abella unchanged. We show that these tacticals can eliminate many commonly encountered kinds of administrative lemmas that would otherwise have to be proven manually, which is a common source of complaints from Abella users.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125362623","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: 8
Idris: Implementing a Dependently Typed Programming Language Idris:实现一种依赖类型的编程语言
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2014-07-17 DOI: 10.1145/2631172.2631174
Edwin C. Brady
{"title":"Idris: Implementing a Dependently Typed Programming Language","authors":"Edwin C. Brady","doi":"10.1145/2631172.2631174","DOIUrl":"https://doi.org/10.1145/2631172.2631174","url":null,"abstract":"Idris is a purely functional programming language with dependent types. As well as supporting theorem proving, Idris is intended to be a general purpose programming language. As such, it provides high-level concepts such as implicit syntax, type classes and \"do\" notation.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123860013","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
Hybrid Extensions in a Logical Framework 逻辑框架中的混合扩展
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2014-07-17 DOI: 10.1145/2631172.2631178
Taus Brock-Nannestad, Nicolas Guenot, Agata Murawska, C. Schürmann
{"title":"Hybrid Extensions in a Logical Framework","authors":"Taus Brock-Nannestad, Nicolas Guenot, Agata Murawska, C. Schürmann","doi":"10.1145/2631172.2631178","DOIUrl":"https://doi.org/10.1145/2631172.2631178","url":null,"abstract":"We discuss the extension of the LF logical framework with operators for manipulating worlds, as found in hybrid logics or in the HLF framework. To overcome the restrictions of HLF, we present a more general approach to worlds in LF, where the structure of worlds can be described in an explicit way. We give a canonical presentation for this system and discuss the encoding of logical systems, beyond the limited scope of linear logic that formed the main goal of HLF.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114621938","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
Internal Adequacy of Bookkeeping in Coq Coq内部记账充分性
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2014-07-17 DOI: 10.1145/2631172.2631180
Alberto Ciaffaglione, Ivan Scagnetto
{"title":"Internal Adequacy of Bookkeeping in Coq","authors":"Alberto Ciaffaglione, Ivan Scagnetto","doi":"10.1145/2631172.2631180","DOIUrl":"https://doi.org/10.1145/2631172.2631180","url":null,"abstract":"We focus on a common problem encountered in encoding and formally reasoning about a wide range of formal systems, namely, the representation of a typing environment. In particular, we apply the bookkeeping technique to a well-known case study (i.e., System F<:'s type language), proving in Coq an internal correspondence with a more standard representation of the typing environment as a list of pairs.\u0000 In order to keep the signature readable and concise, we make use of higher-order abstract syntax (HOAS), which allows us to deal smoothly with the representation of the universal binder of System F<: type language.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127811054","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
A Framework for the Verified Transformation of Functional Programs 函数程序的可验证转换框架
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2014-07-17 DOI: 10.1145/2631172.2631175
G. Nadathur
{"title":"A Framework for the Verified Transformation of Functional Programs","authors":"G. Nadathur","doi":"10.1145/2631172.2631175","DOIUrl":"https://doi.org/10.1145/2631172.2631175","url":null,"abstract":"The compilation of functional programs relies on transformations that simplify their structure while ostensibly preserving their meanings. We argue that the combination of the λProlog language and the Abella interactive theorem-prover provide a natural framework for the verified implementation of such transformations. Underlying this argument is the fact that the transformations are syntax-directed and rule-based, with the important proviso that they pay attention to and also modify the binding structure of programs. The logic of higher-order hereditary Harrop formulas, the HoHH logic for short, is well-suited to formalizing such descriptions especially because of the support it provides for the higher-order representation of syntax. By virtue of the computational interpretation of the HoHH logic embodied in λProlog, these formalizations become implementations of the corresponding transformations. The logic that underlies Abella embeds the HoHH logic and provides a complementary capability for reasoning flexibly and succinctly about the properties of specifications written in the HoHH logic. In this presentation, we will consider typical functional program transformations and show how these twin capabilities can be exploited in their verified implementation; we will especially focus on demonstrating the benefits of a higher-order representation of syntax in both specification and reasoning. We will also discuss an extension to the logic underlying Abella for treating logical relations, a notion that is often needed in semantics preservation arguments.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132054687","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
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