Automatically proving equivalence by type-safe reflection

We are also grateful for the support of the Scottish Informatics and Computer Science Alliance (SICSA) and EPSRC grant EP/N024222/1.One difficulty with reasoning and programming with dependent types is that proof obligations arise naturally once programs become even moderately sized. For example, implementing an adder for binary numbers indexed over their natural number equivalents naturally leads to proof obligations for equalities of expressions over natural numbers. The need for these equality proofs comes, in intensional type theories, from the fact that the propositional equality enables us to prove as equal terms that are not judgementally equal, which means that the typechecker can’t always obtain equalities by reduction. As far as possible, we would like to solve such proof obligations automatically. In this paper, we show one way to automate these proofs by reflection in the dependently typed programming language Idris. We show how defining reflected terms indexed by the original Idris expression allows us to construct and manipulate proofs. We build a hierarchy of tactics for proving equivalences in semi-groups, monoids, commutative monoids, groups, commutative groups, semi-rings and rings. We also show how each tactic reuses those from simpler structures, thus avoiding duplication of code and proofs.Postprin

Revisiting Language Support for Generic Programming: When Genericity Is a Core Design Goal

Context Generic programming, as defined by Stepanov, is a methodology for writing efficient and reusable algorithms by considering only the required properties of their underlying data types and operations. Generic programming has proven to be an effective means of constructing libraries of reusable software components in languages that support it. Generics-related language design choices play a major role in how conducive generic programming is in practice. Inquiry Several mainstream programming languages (e.g. Java and C++) were first created without generics; features to support generic programming were added later, gradually. Much of the existing literature on supporting generic programming focuses thus on retrofitting generic programming into existing languages and identifying related implementation challenges. Is the programming experience significantly better, or different when programming with a language designed for generic programming without limitations from prior language design choices? Approach We examine Magnolia, a language designed to embody generic programming. Magnolia is representative of an approach to language design rooted in algebraic specifications. We repeat a well-known experiment, where we put Magnolia’s generic programming facilities under scrutiny by implementing a subset of the Boost Graph Library, and reflect on our development experience. Knowledge We discover that the idioms identified as key features for supporting Stepanov-style generic programming in the previous studies and work on the topic do not tell a full story. We clarify which of them are more of a means to an end, rather than fundamental features for supporting generic programming. Based on the development experience with Magnolia, we identify variadics as an additional key feature for generic programming and point out limitations and challenges of genericity by property. Grounding Our work uses a well-known framework for evaluating the generic programming facilities of a language from the literature to evaluate the algebraic approach through Magnolia, and we draw comparisons with well-known programming languages. Importance This work gives a fresh perspective on generic programming, and clarifies what are fundamental language properties and their trade-offs when considering supporting Stepanov-style generic programming. The understanding of how to set the ground for generic programming will inform future language design.publishedVersio

Hierarchy Builder: algebraic hierarchies made easy in Coq with Elpi

International audienceIt is nowadays customary to organize libraries of machine checked proofs around hierarchies of algebraic structures [2, 6, 8, 16, 18, 23, 27]. One influential example is the Mathematical Components library on top of which the long and intricate proof of the Odd Order Theorem could be fully formalized [14]. Still, building algebraic hierarchies in a proof assistant such as Coq [9] requires a lot of manual labor and often a deep expertise in the internals of the prover [13, 17]. Moreover, according to our experience [26], making a hierarchy evolve without causing breakage in client code is equally tricky: even a simple refactoring such as splitting a structure into two simpler ones is hard to get right. In this paper we describe HB, a high level language to build hierarchies of algebraic structures and to make these hierarchies evolve without breaking user code. The key concepts are the ones of factory, builder and abbreviation that let the hierarchy developer describe an actual interface for their library. Behind that interface the developer can provide appropriate code to ensure retro compatibility. We implement the HB language in the hierarchy-builder addon for the Coq system using the Elpi [11, 28] extension language

Revisiting Language Support for Generic Programming: When Genericity Is a Core Design Goal

ContextGeneric programming, as defined by Stepanov, is a methodology for writing efficient and reusable algorithms by considering only the required properties of their underlying data types and operations. Generic programming has proven to be an effective means of constructing libraries of reusable software components in languages that support it. Generics-related language design choices play a major role in how conducive generic programming is in practice.InquirySeveral mainstream programming languages (e.g. Java and C++) were first created without generics; features to support generic programming were added later, gradually. Much of the existing literature on supporting generic programming focuses thus on retrofitting generic programming into existing languages and identifying related implementation challenges. Is the programming experience significantly better, or different when programming with a language designed for generic programming without limitations from prior language design choices?ApproachWe examine Magnolia, a language designed to embody generic programming. Magnolia is representative of an approach to language design rooted in algebraic specifications. We repeat a well-known experiment, where we put Magnolia’s generic programming facilities under scrutiny by implementing a subset of the Boost Graph Library, and reflect on our development experience.KnowledgeWe discover that the idioms identified as key features for supporting Stepanov-style generic programming in the previous studies and work on the topic do not tell a full story. We clarify which of them are more of a means to an end, rather than fundamental features for supporting generic programming. Based on the development experience with Magnolia, we identify variadics as an additional key feature for generic programming and point out limitations and challenges of genericity by property.GroundingOur work uses a well-known framework for evaluating the generic programming facilities of a language from the literature to evaluate the algebraic approach through Magnolia, and we draw comparisons with well-known programming languages.ImportanceThis work gives a fresh perspective on generic programming, and clarifies what are fundamental language properties and their trade-offs when considering supporting Stepanov-style generic programming. The understanding of how to set the ground for generic programming will inform future language design.</p

Type-Theoretic Signatures for Algebraic Theories and Inductive Types

We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. We describe syntax and semantics for three classes of algebraic theories: finitary quotient inductive-inductive theories, their infinitary generalization, and finally higher inductive-inductive theories. In each case, an algebraic signature is a typing context or a closed type in a specific type theory