This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtyping relation over the underlying types. Among the basic types is the intersection type which plays a critical role in the applications because it provides a method of composing program components. The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It is also used to define communications protocols as infinite state automata. The article illustrates the role of these formal automata in defining the services of a distributed group communications system. In both applications the inheritance mechanisms allow reuse of proofs and the statement of general properties of system composition. 1

In this paper we give the first example of a significant piece of formal mathematics conducted in a hybrid of two different interactive systems. We constructively prove a theorem in Nuprl, from which a program can be extracted, but we use classical mathematics imported from HOL, and a connection to some of HOL's definitional packages, for parts of the proof that do not contribute to the program.

Abstract. We present a semantically correct embedding of a subset of the Ocaml programming language into the type theory of NuPRL. The subset is that needed to build the Ensemble group communication system. We describe the essential methodologies for representing language constructs by type-theoretical expressions. Tactics representing derived inference rules and a programming logic for these constructs will be discussed as well as algorithms for translating an Ocaml-program into NuPRL-objects and vice versa. The formal representations and the translation algorithms will serve as the foundation for the development of automated reasoning tools for the

We present a constructive formalization of the Myhill-Nerode theorem on the minimization of finite automata that follows the account in Hopcroft and Ullman's book Formal Languages and Their Relation to Automata. We chose to formalize this theorem because it illustrates many points critical to formalization of computational mathematics, especially the extraction of an important algorithm from a proof as a method of knowing that the algorithm is correct. It also gave us an opportunity to experiment with a constructive implementation of quotient sets. We carried out the formalization in Nuprl, an interactive theorem prover based on constructive type theory. Nuprl borrows an implementation of the ML language from the LCF system of Milner, Gordon, and Wadsworth, and makes heavy use of the notion of tactic pioneered by Milner in LCF. We are interested in the pedagogical value of electronic formal mathematical texts and have put our formalization on the World Wide Web. Readers are invited to ...

Isabelle and NuPRL are two theorem proving environments that are written in different dialects of ML using different formula syntaxes and different logical foundations. In spite of this, they have similar sets of basic theories, representing the same mathematical concepts.

. We give a new semantic foundation for type theories in the lineage of Martin-Lof's "polymorphic extensional" type theory, and use it to give a model of the constructive type theory of the interactive theorem proving system Nuprl. These type theories are based on an operational semantics of an untyped programming language. We show how to integrate classical set-theoretic objects, such as functions-as-graphs and equivalence classes, into this operational framework. The new semantics is dramatically simpler than the previous ones, and enables direct reasoning about classical mathematics. A practical consequence is that it justifies a useful embedding of the logic of the HOL theorem prover that gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the years. 1 Introduction The so-called "polymorphic extensional" type theory of Martin-Lof (Martin-Lof, 1982) has two features that set it apart from other constructive type t...

We describe a generic switching protocol for the construction of hybrid protocols and prove it correct with the Nuprl proof development system. For this purpose we introduce the concept of metaproperties and use them to formally characterize communication properties that can be preserved by switching. We also identify switching invariants that an implementation of the switching protocol must satisfy in order to work correctly.

This paper presents a rigorous, uni ed framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client interface is linked to a server interface by an asymmetric connection consisting of a pair of translations. Answers to requests are trustable in the sense that they are correct provided a small set of prescribed conditions are satis ed. The framework is robust with respect to interface extension and can process requests for abstract services, where the server interface is not fully speci ed.

We describe a generic switching protocol for the construction of hybrid protocols and prove it correct with the Nuprl proof development system. We introduce the concept of meta-properties to characterize communication properties that can be preserved by switching and identify switching invariants that an implementation of the switching protocol must satisfy in order to work correctly. Our work shows how a theorem prover with a rich specification language can contribute to the design and implementation of verifiably correct adaptive protocols and that it can have a large impact when being engaged at the

The basic concepts of type theory are fundamental to computer science, logic and mathematics. Indeed, the language of type theory connects these regions of science. It plays a role in computing and information science akin to that of set theory in pure mathematics. There are many excellent accounts of the basic ideas of type theory, especially at the interface of computer science and logic — specifically, in the literature of programming languages, semantics, formal methods and automated reasoning. Most of these are very technical, dense with formulas, inference rules, and computation rules. Here we follow the example of the mathematician Paul Halmos, who in 1960 wrote a 104-page book called Naïve Set Theory intended to make the subject accessible to practicing mathematicians. His book served many generations well. This article follows the spirit of Halmos ’ book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the preface to Halmos ’ book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type theory; the disagreement begins in trying to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced computer science the basic type theoretic facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist eager to study programming languages, or database systems, or computational complexity theory, or distributed systems or information discovery. In type theory, “naïve ” and “formal ” are contrasting words. The present treatment might best be described as informal type theory from a naïve point of view. The concepts are very general and very abstract; therefore they may