The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive 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 ...

We give two nite axiomatizations of indexed inductive-recursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductive-recursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductive-recursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.

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This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type theory and how to build initial categories with families. Initial categories with families will be term models which play the role of canonical syntax for dependent type theory. We also discuss the formalization of the result that categories with finite limits give rise to categories with families. This yields a type-theoretic perspective on Curien’s work on “substitution up to isomorphism”. Our formalization is being carried out in the proof assistant Agda 2 developed at Chalmers. 1

We discuss different ways to represent the syntax of dependent types using Martin-Lof type theory as a metalanguage. In particular, we show how to give an intrinsic syntax in which meaningful contexts, types in a context, and terms of a certain type in a context, are generated directly without first introducing raw terms, types, and contexts. In the first representation we define inductively the normal contexts, types, and terms of the pure theory of dependent types. Simultaneously substitution and lifting are defined recursively. Equality in the object language is here syntactic equality and is represented by the equality of the metalanguage. The second representation is a calculus of explicit substitutions. This is a pure inductive definition and proofs of equalities are generated simultaneously. As for Curien's explicit syntax there are term constructors corresponding to applications of the type equality rules, and coherence conditions related to those appearing in category theory. ...