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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 re ..."
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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 50 (17 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.
Structured Type Theory
, 1999
"... Introduction We present our implementation AGDA of type theory. We limit ourselves in this presentation to a rather primitive form of type theory (dependent product with a simple notion of sorts) that we extend to structure facility we find in most programming language: let expressions (local defin ..."
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Cited by 41 (4 self)
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Introduction We present our implementation AGDA of type theory. We limit ourselves in this presentation to a rather primitive form of type theory (dependent product with a simple notion of sorts) that we extend to structure facility we find in most programming language: let expressions (local definition) and a package mechanism. We call this language Structured Type Theory. The first part describes the syntax of the language and an informal description of the typechecking. The second part contains a detailed description of a core language, which is used to implement Strutured Type Theory. We give a realisability semantics, and typechecking rules are proved correct with respect to this semantics. The notion of metavariables is explained at this level. The third part explains how to interpret Structured Type Theory in this core language. The main contributions are: ffl use of explicit substitution to simplify and make
Type Theory with FirstOrder Data Types and SizeChange Termination
, 2004
"... We prove normalization for a dependently typed lambdacalculus extended with firstorder data types and computation schemata for firstorder sizechange terminating recursive functions. Sizechange termination, introduced by C.S. Lee, N.D. Jones and A.M. BenAmram, can be seen as a generalized form ..."
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We prove normalization for a dependently typed lambdacalculus extended with firstorder data types and computation schemata for firstorder sizechange terminating recursive functions. Sizechange termination, introduced by C.S. Lee, N.D. Jones and A.M. BenAmram, can be seen as a generalized form of structural induction, which allows inductive computations and proofs to be defined in a straightforward manner. The language can be used as a proof system—an extension of MartinLöf’s Logical Framework.
Indexed InductionRecursion
"... An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive ..."
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An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRDs in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRDs includes essentially all definitions of sets which occur in MartinLöf type theory. We show in particular that MartinLöf’s computability predicates for dependent types and Palmgren’s higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions. We give two axiomatisations. The first and more restricted one formalises a principle for introducing meaningful IIRD by using the dataconstruct in the original version of the proof assistant Agda for MartinLöf type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the restricted one. If we add an extensional identity relation to our logical framework, we show that the theories of restricted and general IIRD are equivalent by interpreting them in each other. Finally, we show the consistency of our theories by constructing a model in classical set theory extended by a Mahlo cardinal.
An Implementation of Type:Type
, 2000
"... We present a denotational semantics of a type system with dependent types, where types are interpreted as finitary projections. We prove then the correctness of a typechecking algorithm w.r.t. this semantics. In this way, we can justify some simple optimisation in this algorithm. We then sketch h ..."
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We present a denotational semantics of a type system with dependent types, where types are interpreted as finitary projections. We prove then the correctness of a typechecking algorithm w.r.t. this semantics. In this way, we can justify some simple optimisation in this algorithm. We then sketch how to extend this semantics to allow a simple record mechanism with manifest fields.