Characteristics of de Bruijn's early proof checker Automath

TL;DR

This paper explores the syntax and foundational aspects of Automath, an early theorem prover, and relates it to the calculus of constructions, highlighting its influence on modern proof systems like Coq.

Contribution

It provides a detailed analysis of Automath's syntax, relating it to the calculus of constructions, and introduces a generic Automath system within a modern syntactic framework.

Findings

Automath's syntax is based on a derivation system similar to CC.

A generic Automath system encapsulating common versions is presented.

The system uses λD, an extension of CC with definitions.

Abstract

The `mathematical language' Automath, conceived by N.G. de Bruijn in 1968, was the first theorem prover actually working and was used for checking many specimina of mathematical content. Its goals and syntactic ideas inspired Th. Coquand and G. Huet to develop the calculus of constructions, CC, which was one of the first widely used interactive theorem provers and forms the basis for the widely used Coq system. The original syntax of Automath is not easy to grasp. Yet, it is essentially based on a derivation system that is similar to the Calculus of Constructions (`CC'). The relation between the Automath syntax and CC has not yet been sufficiently described, although there are many references in the type theory community to Automath. In this paper we focus on the backgrounds and on some uncommon aspects of the syntax of Automath. We expose the fundamental aspects of a `generic' Automath system, encapsulating the most common versions of Automath. We present this generic Automath system in a modern syntactic frame. The obtained system makes use of {\lambda}D, a direct extension of CC with definitions.