Journal of Symbolic Logic

A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory

Peter Dybjer

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The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Lof's universe a la Tarski. A set U$_0$ of codes for small sets is generated inductively at the same time as a function T$_0$, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U$_0$ are generated. In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is implicit in Martin-Lof's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

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J. Symbolic Logic, Volume 65, Issue 2 (2000), 525-549.

First available in Project Euclid: 6 July 2007

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Dybjer, Peter. A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory. J. Symbolic Logic 65 (2000), no. 2, 525--549.

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