A general formulation of simultaneous inductive-recursive definitions in type theory

Journal of Symbolic Logic 65 (2):525-549 (2000)
Abstract
The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe á 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-Löf'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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DOI 10.2307/2586554
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References found in this work BETA
Propositional Functions and Families of Types.Jan M. Smith - 1989 - Notre Dame Journal of Formal Logic 30 (3):442-458.

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Citations of this work BETA
Regular Universes and Formal Spaces.Erik Palmgren - 2006 - Annals of Pure and Applied Logic 137 (1):299-316.
Induction–Recursion and Initial Algebras.Peter Dybjer & Anton Setzer - 2003 - Annals of Pure and Applied Logic 124 (1-3):1-47.

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