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Partiality and recursion in interactive theorem provers – an overview

Published online by Cambridge University Press:  10 November 2014

ANA BOVE ,
ALEXANDER KRAUSS  and
MATTHIEU SOZEAU
ANA BOVE
Affiliation:
Department of Computer Science and Engineering, Chalmers University of Technology, 412 96, Gothenburg, Sweden Email: bove@chalmers.se
ALEXANDER KRAUSS
Affiliation:
Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany Email: krauss@in.tum.de
MATTHIEU SOZEAU
Affiliation:
INRIA Paris, 23 avenue d'Italie, 75013 Paris, France Email: matthieu.sozeau@inria.fr
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Abstract

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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Information
Mathematical Structures in Computer Science , Volume 26 , Special Issue 1: Special Issue: Dependently Typed Programming , January 2016 , pp. 38 - 88
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Copyright © Cambridge University Press 2014 

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