A Few Constructions on Constructors

McBride, Conor; Goguen, Healfdene; McKinna, James

doi:10.1007/11617990_12

Cited by 19 publications

(8 citation statements)

References 20 publications

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“…Behind the scenes, these are "compiled" down into eliminators and auxiliary definitions automatically generated by Lean whenever we declare an inductive family. This compiler is based on ideas from [13,9,21,4]. The default compilation method supports structural recursion, i.e.…”

Section: Elaborationmentioning

confidence: 99%

The Lean Theorem Prover (System Description)

Moura

Kong

Avigad

et al. 2015

Lecture Notes in Computer Science

262

147

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Abstract. Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

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Section: Elaborationmentioning

confidence: 99%

The Lean Theorem Prover (System Description)

Moura

Kong

Avigad

et al. 2015

Lecture Notes in Computer Science

262

147

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show abstract

“…Furthermore, they simplify implementation of the evaluator and provide easy optimisation opportunities (Brady et al, 2003). However, it requires the implementation of extra machinery for constructor manipulation (McBride et al, 2006) and so we have avoided it in the present implementation.…”

Section: Related Workmentioning

confidence: 99%

Idris, a general-purpose dependently typed programming language: Design and implementation

Brady¹

2013

J. Funct. Prog.

248

210

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Many components of a dependently typed programming language are by now well understood, for example, the underlying type theory, type checking, unification and evaluation. How to combine these components into a realistic and usable high-level language is, however, folklore, discovered anew by successive language implementors. In this paper, I describe the implementation of Idris, a new dependently typed functional programming language. Idris is intended to be a general-purpose programming language and as such provides high-level concepts such as implicit syntax, type classes and do notation. I describe the high-level language and the underlying type theory, and present a tactic-based method for elaborating concrete high-level syntax with implicit arguments and type classes into a fully explicit type theory. Furthermore, I show how this method facilitates the implementation of new high-level language constructs.

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“…Before we begin this construction, we repeat some standard definitions from type theory (Section 4), including telescopic equality. We then recall some standard equipment for inductive datatypes given by McBride et al (2006): case analysis, structural recursion, no confusion, and acyclicity, of which the latter two are slightly adapted to take the additional dependencies on equality proofs into account (Section 5). This is a return to an early version of McBride (1998) that was still based on homogeneous equality.…”

Section: Soundnessmentioning

confidence: 99%

Eliminating dependent pattern matching without K

Cockx¹,

Devriese²,

Piessens³

2016

J. Funct. Prog.

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Dependent pattern matching is an intuitive way to write programs and proofs in dependently typed languages. It is reminiscent of both pattern matching in functional languages and case analysis in on-paper mathematics. However, in general, it is incompatible with new type theories such as homotopy type theory (HoTT). As a consequence, proofs in such theories are typically harder to write and to understand. The source of this incompatibility is the reliance of dependent pattern matching on the so-called K axiom – also known as the uniqueness of identity proofs – which is inadmissible in HoTT. In this paper, we propose a new criterion for dependent pattern matching without K, and prove it correct by a translation to eliminators in the style of Goguen et al. (2006 Algebra, Meaning, and Computation). Our criterion is both less restrictive than existing proposals, and solves a previously undetected problem in the old criterion offered by Agda. It has been implemented in Agda and is the first to be supported by a formal proof. Thus, it brings the benefits of dependent pattern matching to contexts where we cannot assume K, such as HoTT.

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A Few Constructions on Constructors

Cited by 19 publications

References 20 publications

The Lean Theorem Prover (System Description)

The Lean Theorem Prover (System Description)

Idris, a general-purpose dependently typed programming language: Design and implementation

Eliminating dependent pattern matching without K

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