Equations: A Dependent Pattern-Matching Compiler

Sozeau, Matthieu

doi:10.1007/978-3-642-14052-5_29

Cited by 33 publications

(34 citation statements)

References 11 publications

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“…Though some recent work has shown how to add Agda-style pattern matching to Coq, this is still only available as an experimental language extension [27]. Additionally, developing in Agda allowed us to deal with non-termination more conveniently-while Coq must be able to see that a definition terminates before moving on, Agda shows the user where it can not prove termination and allows other work to continue.…”

Section: Discussionmentioning

confidence: 99%

Generic Programming with Dependent Types

Weirich

Casinghino

2012

Lecture Notes in Computer Science

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Abstract. Some programs are doubly generic. For example, map is datatype-generic in that many different data structures support a mapping operation. A generic programming language like Generic Haskell can use a single definition to generate map for each type. However, map is also arity-generic because it belongs to a family of related operations that differ in the number of arguments. For lists, this family includes familiar functions from the Haskell standard library (such as repeat, map, and zipWith).This tutorial explores these forms of genericity individually and together. These two axes are not orthogonal: datatype-generic versions of repeat, map and zipWith have different arities of kind-indexed types. We explore these forms of genericity in the context of the Agda programming language, using the expressiveness of dependent types to capture both forms of genericity in a common framework. Therefore, this tutorial serves as an introduction to dependently typed languages as well as generic programming.The target audience of this work is someone who is familiar with functional programming languages, such as Haskell or ML, but would like to learn about dependently typed languages. We do not assume prior experience with Agda, type-or arity-generic programming.

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Section: Discussionmentioning

confidence: 99%

Generic Programming with Dependent Types

Weirich

Casinghino

2012

Lecture Notes in Computer Science

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“…Equations. The Equations package (Sozeau 2010) extends the support of the Coq system to perform dependent pattern-matching and well-founded recursion on inductive families. Moreover, it provides auxiliary lemmas such as the recursive equations of the function and an induction principle, much like Function.…”

Section: The Agda2atp Tool In Agdamentioning

confidence: 99%

Partiality and recursion in interactive theorem provers – an overview

Bove¹,

Krauß²,

Sozeau³

2014

Math. Struct. Comp. Sci.

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The use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalizing mathematics is becoming more common and feasible in practice. However, most mature theorem provers lack a direct treatment of partial and general recursive functions; overcoming this weakness has been the objective of intensive research during the last decades. In this article, we review several techniques that have been proposed in the literature to simplify the formalization of partial and general recursive functions in interactive theorem provers. Moreover, we classify the techniques according to their theoretical basis and their practical use. This uniform presentation of the different techniques facilitates the comparison and highlights their commonalities and differences, as well as their relative advantages and limitations. We focus on theorem provers based on constructive type theory (in particular, Agda and Coq) and higher-order logic (in particular Isabelle/HOL). Other systems and logics are covered to a certain extent, but not exhaustively. In addition to the description of the techniques, we also demonstrate tools which facilitate working with the problematic functions in particular theorem provers.

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“…Using this representation, lambdas and other binders can be seen as terms with holes, which can be filled with other terms (typically values). We use the Coq Equations library [Sozeau 2010] to define the reducibility logical relation v . This library facilitates the use of functions which are defined recursively based on a well-founded measure.…”

Section: Formalization In the Coq Proof Assistantmentioning

confidence: 99%

System FR: formalized foundations for the stainless verifier

Hamza

Voirol

Kunčak

2019

Proc. ACM Program. Lang.

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We present the design, implementation, and foundation of a verifier for higher-order functional programs with generics and recursive data types. Our system supports proving safety and termination using preconditions, postconditions and assertions. It supports writing proof hints using assertions and recursive calls. To formalize the soundness of the system we introduce System FR, a calculus supporting System F polymorphism, dependent refinement types, and recursive types (including recursion through contravariant positions of function types). Through the use of sized types, System FR supports reasoning about termination of lazy data structures such as streams. We formalize a reducibility argument using the Coq proof assistant and prove the soundness of a type-checker with respect to call-by-value semantics, ensuring type safety and normalization for typeable programs. Our program verifier is implemented as an alternative verification-condition generator for the Stainless tool, which relies on the Inox SMT-based solver backend for automation. We demonstrate the efficiency of our approach by verifying a collection of higher-order functional programs comprising around 14000 lines of polymorphic higher-order Scala code, including graph search algorithms, basic number theory, monad laws, functional data structures, and assignments from popular Functional Programming MOOCs.

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Equations: A Dependent Pattern-Matching Compiler

Cited by 33 publications

References 11 publications

Generic Programming with Dependent Types

Generic Programming with Dependent Types

Partiality and recursion in interactive theorem provers – an overview

System FR: formalized foundations for the stainless verifier

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