A general formulation of simultaneous inductive-recursive definitions in type theory

Abstract

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 ...

Viewing λ-terms through maps

“…Since divisibility can also be viewed as a boolean-valued function defined by recursion over L, Fig. 2 could be reformulated as a simultaneous inductive-recursive definition (Dybjer [6]). Further, since the relation m | M is decidable, one can consider a formalization where the constructor mask only takes the first two arguments, and the third (proof) argument is irrelevant.…”

Small Induction Recursion

“…Dybjer and Setzer developed the theory of induction recursion to cover exactly such inductive definitions where the indices and the data must be defined simultaneously. The first presentation of induction-recursion [8] was as an external schema. In later presentations, inductive recursive definitions are given via a type of codes IR I O.…”

Typed Applicative Structures and Normalization by Evaluation for System F ω

“…What is the challenge when stepping up to impredicativity? Predicative type theories allow to define the semantics of types from below via induction-recursion [Dyb00], and the reification function can be defined by induction on types. This fails in the presence of impredicativity, where one first has to lay out a lattice of semantic type candidates and then define impredicative quantification using an intersection over all candidates [GLT89].…”