Inproceedings,
Inductive Sets and Families in Martin-Löf's Type Theory and Their Set-Theoretic Semantics
Logical Frameworks, page 280--306. Cambridge University Press, (1991)
Abstract
Martin-Lof's type theory is presented in several steps. The kernel is a dependently typed -calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. At each step syntax, inference rules, and set-theoretic semantics are given. 1 Introduction Usually Martin-Lof's type theory is presented as a closed system with rules for a finite collection of set formers. But it is also often pointed out that the system is in principle open to extension: we may introduce new sets when there is a need for them. The principle is that a set is by definition inductively generated - it is defined by its introduction rules, which are rules for generating its elements. The elimination rule is determined by the introduction rules and expresses definition by primitive recursion on the way the elements of the set are generated. (In this paper I shall use the term ...
