**
RECENT RESULTS IN TYPE THEORY AND
THEIR RELATIONSHIP TO AUTOMATH **

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by Robert L. Constable

Published: Fairouz Kamareddine (eds.),

Thirty Five Years of Automating Mathematics 1–11.

©
2003, Kluwer Academic Publishers. Printed in the Netherlands.

**About**

The notion of a telescope is basic to Automath’s theory structure; telescopes provide the context for theorems. A dependent record type is an internal version of a telescope and is used in Nuprl to define theories. This paper shows how A. Kopylov defines these record types in terms of dependent intersections, a new type constructor.

Definitional equality and book equality are fundamental concepts basic to Automath. In computational type theories these concepts appear in a different form, as computational equalities and as quotient types. Questions about these concepts have led to interesting discoveries about types and open problems. This paper presents a new formulation of quotient types by A. Nogin, and an open question about them.