Inductively defined types in the Calculus of Constructions

Abstract

We dene the notion of an inductively dened type in the Calculus of Constructions and show how inductively dened types can be represented by closed types. We show that all primitive recursive functionals over these inductively dened types are also representable. This generalizes work by B ohm & Berarducci on synthesis of functions on term algebras in the second-order polymorphic -calculus (F 2 ). We give several applications of this generalization, including a representation of F 2 -programs in F 3 , along with a denition of functions reify, reflect, and eval for F 2 in F 3 . We also show how to dene induction over inductively dened types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, F ! . This is because a proof by induction can be realized by primitive recursion, which is already denable in F ! .

Recursion on Nested Datatypes in Dependent Type Theory

“…In these cases, our proposal is more generic but offers less comfort since it has neither advanced pattern matching nor guardedness checking. PList and Lam are strictly positive, but Bush and LamE are not even considered to be positive [19] (see [14] for a notion based on polarity that covers these examples). Since there was no system that combined the termination guarantee for recursion schemes on truly nested datatypes with a logic to reason about the defined functions, it seems only natural that examples like Bush and LamE did not receive more attention.…”

LEAP: A language with eval and polymorphism

“…This is an extension of the datatype construction in ML, since constructors may be explicitly polymorphic. It is shown in [13] (extending ideas of B ohm & Berarducci [1]) that these types do not require an addition to the core language, since inductively dened types are representable by closed types. The basic problem is to be able to explicitly dene a function from types to types, such that is a type representing programs of type .…”

“…This is a special case of a very general transformation from an inductive denition of a data type into an encoding into F ! described in [13]. The denitions of the constructors in this encoding can be found in Figure 1.…”

“…Let us rst present the function in the form of an iterative denition (see [1] for a discussion of iterative denitions in F 2 and [13] for a generalization that encompasses F ! ).…”

“…An example of such a specication appears in Section 5.1, where we dene the type using this syntactic extension. A full discussion of the denition of primitive recursion and inductively-dened data types in Pure LEAP is given in another paper [13].…”