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A General Formulation of Simultaneous InductiveRecursive Definitions in Type Theory
 Journal of Symbolic Logic
, 1998
"... The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf'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 recursi ..."
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Cited by 65 (10 self)
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The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf'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 MartinLöf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous inductionrecursion. 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 ...
Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
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Cited by 44 (16 self)
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We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive 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 inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
Normalization by evaluation for MartinLöf type theory with one universe
 IN 23RD CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, MFPS XXIII, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2007
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Formalizing categorical models of type theory in type theory
 In International Workshop on Logical Frameworks and MetaLanguages: Theory and Practice
, 2007
"... 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 de ..."
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Cited by 5 (2 self)
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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 typetheoretic 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
On the Syntax of Dependent Types and the Coherence Problem (working draft)
, 1994
"... We discuss different ways to represent the syntax of dependent types using MartinLof 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 ..."
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We discuss different ways to represent the syntax of dependent types using MartinLof 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. ...