# Selected Papers on Automath

**Edited by**

- R.P. Nederpelt, Eindhoven University of Technology, The Netherlands
- J.H. Geuvers, Eindhoven University of Technology, The Netherlands
- R.C. de Vrijer, Free University, Department of Mathematics and Informatics, Amsterdam, The Netherlands

The present volume contains a considered choice of the existing literature on Automath. Many of the papers included in the book have been published in journals or conference proceedings, but a number have only circulated as research reports or have remained unpublished. The aim of the editors is to present a representative selection of existing articles and reports and of material contained in dissertations, giving a compact and more or less complete overview of the work that has been done in the Automath research field, from the beginning to the present day. Six different areas have been distinguished, which correspond to Parts A to F of the book. These areas range from general ideas and motivation, to detailed syntactical investigations.

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### Book information

- Published: October 1994
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-89822-7

### Table of Contents

**Part A: Motivation and Exposition.** Verification of mathematical proofs by a computer (N.G. de Bruijn). The mathematical language Automath, its usage, and some of its extensions (N.G. de Bruijn). A description of Automath and some aspects of its language theory (D.T. van Daalen). Formalization of classical mathematics in Automath (J. Zucker). A survey of the project Automath (N.G. de Bruijn). The language theory of Automath. Chapter I, Sections 1-5 (Introduction) (D.T. van Daalen). Reflections on Automath (N.G. de Bruijn). Type systems - basic ideas and applications (R.P. Nederpelt). **Part B: Language Definition and Special Subjects.** Description of AUT-68 (L.S. van Benthem Jutting). AUT-SL, a single line version of Automath (N.G. de Bruijn). Some extensions of Automath: The AUT-4 family (N.G. de Bruijn). AUT-QE without type inclusion (N.G. de Bruijn). Checking Landau's Grundlagen

in the Automath system. Appendix 9 (AUT-SYNT) (L.S. van Benthem Jutting). The language theory of Automath. Chapter VIII, 1 and 2 (AUT-II) (D.T. van Daalen). Generalizing Automath by means of a lambda-typed lambda calculus (N.G. de Bruijn). Lambda calculus extended with segments. Chapter 1, Sections 1.1 and 1.2 (Introduction) (H. Balsters). **Part C: Theory.** A normal form theorem in a &lgr;-calculus with types (L.S. van Benthem Jutting). Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem (N.G. de Bruijn). Strong normalization in a typed lambda calculus with lambda structured types (R.P. Nederpelt). Big trees in a &lgr;-calculus with &lgr;-expressions as types (R.C. de Vrijer). The language theory of Automath. Parts of Chapters II, IV, V-VIII (D.T. van Daalen). The language theory of &Lgr;∞, a typed &lgr;-calculus where terms are typed (L.S. van Benthem Jutting). **Part D: Text Examples.** Example of a text written in Automath (N.G. de Bruijn). Checking Landau's Grundlagen

in the Automath system. Parts of Chapters 0, 1 and 2 (Introduction, Preparation, Translation) (L.S. van Benthem Jutting). Checking Landau's Grundlagen

in the Automath system. Chapter 4 (Conclusions) (L.S. van Benthem Jutting). A text fragment from Zucker's Real Analysis

(L.S. van Benthem Jutting, R.C. de Vrijer). Checking Landau's Grundlagen

in the Automath system. Appendices 3 and 4 (The PN-lines; Excerpt for Satz 27

(L.S. van Benthem Jutting). **Part E: Verification.** A verifying program for Automath (I. Zandleven). Checking Landau's Grundlagen

in the Automath system. Parts of Chapter 3 (Verification) (L.S. van Benthem Jutting). An implementation of substitution in a &lgr;-calculus with dependent types (L.S. van Benthem Jutting). **Part F: Related Topics.** Set theory with type restrictions (N.G. de Bruijn). Formalization of constructivity in Automath (N.G. de Bruijn). The Mathematical Vernacular, a language for mathematics with typed sets (N.G. de Bruijn). Relational semantics in an integrated system (R.M.A. Wieringa). Computer program semantics in space and time (N.G. de Bruijn). Bibliography. Indexes.