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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. Abstract. The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-LUf's universe a la Tarski. A set Uo of codes for small sets is generated inductively at the same time as a function To, which maps a code to the corresponding small set, is defined by recursion on the way the elements of Uo are generated.In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is implicit in Martin-LUf'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 in the style of Allen.'---P (m > 0), where by does not have any occurrences of P. It is also clear what the appropriate notion of structural recursion is for such a recursive datatype. Note that we allow generalised inductive definitions since a constructor can have functional arguments (m > 0). Therefore, the informal semantic notions of well-founded element and terminating function depend on each other.The introduction of dependent types dramatically increases the expressiveness of the language. In particular, we can interpret intuitionistic predicate logic by following Curry, Howard, and de Bruijn and identify propositions and sets. In addition to the ordinary non-dependent set former 0, 1, +, x, and -?, which can be used for interpreting the logical connectives I, T, V, A, and D, we now also have Y and H, the disjoint union and Cartesian product of a family of sets, which can be used for interpreting the quantifiers 3 and V.However, the appropriate notion corresponding to "strict positivity" becomes more complex in the context of dependent types. Instead of formulating such a general condition for inductive definitions of sets Martin-Ldf [32, 28, 29, 30j gave rules for a collection of specific set formers. However, this collection may be extended when there is a need for it provided the informal semantic principles of the theory are respected. The possibility of formulating a general schema was however mentioned in Martin-Ldf 1972 [32]:The type N is just the prime example of a type introduced by an ordinary inductive definition. However, it seems preferable ...

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A general formulation of inductive and recursive definitions in Martin-Löf's type theory is presented. It extends Backhouse's ‘Do-It-Yourself Type Theory’ to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalisation to type theory of Martin-Löf's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalised induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applications by showing several basic examples: predicate logic, generalised induction, and a formalisation of the untyped lambda calculus.

Martin-L of'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.

We solve the decision problem for simply typed lambda calculus with strong binary sums, equivalently the word problem for free cartesian closed categories with binary coproducts. Our method is based on the semantical technique known as "normalization by evaluation" and involves inverting the interpretation of the syntax into a suitable sheaf model and from this extracting appropriate unique normal forms. There is no rewriting theory involved, and the proof is completely constructive, allowing program extraction from the proof.

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf's type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal.

The traditional notions of strong and weak normalization refer to properties of a binary reduction relation. In this paper we explore an alternative approach to normalization, where we bypass the reduction relation and instead focus on the normalization function, that is, the function which maps a term into its normal form. We work in an intuitionistic metalanguage, and characterize a normalization function as an algorithm which picks a canonical representative from the equivalence class of convertible terms. Hence we also get a decision algorithm for convertibility. Such a normalization function can be constructed by building an appropriate model and a function \quote" which inverts the interpretation function. The normalization function is then obtained by composing the quote function with the interpretation function. We also discuss how to get a simple proof of the property that constructors are one-to-one, which usually is obtained as a corollary of Church-Rosser and normalization in the traditional sense. We illustrate this approach by showing how a glueing model (closely related to the glueing construction used in category theory) gives rise to a normalization algorithm for a combinatory formulation of G odel System T. We then show how the method extends in a straightforward way when we add cartesian products and disjoint unions (full intuitionistic propositional logic under a Curry-Howard interpretation) and trans nite inductive types such as the Brouwer ordinals.