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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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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.
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.