scholarly journals Equality in computer proof-assistants

2015 ◽  
Author(s):  
Adam Grabowski ◽  
Artur Kornilowicz ◽  
Christoph Schwarzweller
Keyword(s):  
Proof Assistants ◽  
Computer Proof

scholarly journals Homotopy theoretic models of identity types

2009 ◽  
Vol 146 (1) ◽  
pp. 45-55 ◽  
Author(s):  
STEVE AWODEY ◽  
MICHAEL A. WARREN
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Theoretical Basis ◽  
Homotopy Theory ◽  
Model Category ◽  
Proof Assistants ◽  
Model Categories ◽  
Wide Range ◽  
Abstract Framework ◽  
Computer Proof

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11,12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such asCoqandAgda(cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

Can a computer proof be elegant?

2001 ◽  
Vol 32 (1) ◽  
pp. 111-114
Author(s):  
Steve Seiden
Keyword(s):  
Computer Proof

scholarly journals CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

2011 ◽  
Vol 21 (4) ◽  
pp. 827-859 ◽  
Author(s):  
FRÉDÉRIC BLANQUI ◽  
ADAM KOPROWSKI
Keyword(s):  
Term Rewriting ◽  
Automated Verification ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Active Subject ◽  
Turing Complete

Termination is an important property of programs, and is notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting. Over the years, many methods and tools have been developed to address the problem of deciding termination for specific problems (since it is undecidable in general). Ensuring the reliability of those tools is therefore an important issue.In this paper we present a library formalising important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools.The sources are freely available athttp://color.inria.fr/.

Design of Point-and-Click User Interfaces for Proof Assistants

2019 ◽  
pp. 86-103
Author(s):  
Bohua Zhan ◽  
Zhenyan Ji ◽  
Wenfan Zhou ◽  
Chaozhu Xiang ◽  
Jie Hou ◽  
...  
Keyword(s):  
User Interfaces ◽  
Proof Assistants

scholarly journals Translating Higher-Order Specifications to Coq Libraries Supporting Hybrid Proofs

10.29007/jqtz ◽  
2018 ◽  
Author(s):  
Nada Habli ◽  
Amy P. Felty
Keyword(s):  
Programming Languages ◽  
Automatic Generation ◽  
Higher Order ◽  
Specification Language ◽  
Proof Assistants ◽  
Abstract Syntax ◽  
Formal Systems ◽  
Ongoing Work ◽  
General Specification ◽  
Automated Proof

We describe ongoing work on building an environment to support reasoning in proof assistants that represent formal systems using higher-order abstract syntax (HOAS). We use a simple and general specification language whose syntax supports HOAS. Using this language, we can encode the syntax and inference rules of a variety of formal systems, such as programming languages and logics. We describe our tool, implemented in OCaml, which parses this syntax, and translates it to a Coq library that includes definitions and hints for aiding automated proof in the Hybrid system. Hybrid itself is implemented in Coq, and designed specifically to reason about such formal systems. Given an input specification, the library that is automatically generated by our tool imports the general Hybrid library and adds definitions and hints for aiding automated proof in Hybrid about the specific programming language or logic defined in the specification. This work is part of a larger project to compare reasoning in systems supporting HOAS. Our current work focuses on Hybrid, Abella, Twelf, and Beluga, and the specification language is designed to be general enough to allow the automatic generation of libraries for all of these systems from a single specification.

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