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NuPRL’s class theory and its applications
 Foundations of Secure Computation, NATO ASI Series, Series F: Computer & System Sciences
, 2000
"... 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 und ..."
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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
The structure of nuprl’s type theory
, 1997
"... on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) ..."
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Cited by 9 (3 self)
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on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)
Näıve computational type theory
 Proof and SystemReliability, Proceedings of International Summer School Marktoberdorf, July 24 to August 5, 2001, volume 62 of NATO Science Series III
, 2002
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Congruence Types
 Proceedings of CSL'95
, 1996
"... . We introduce a typetheoretical framework in which canonical term rewriting systems can be represented faithfully both from the logical and the computational points of view. The framework is based on congruence types, a new syntax which combines inductive, algebraic and quotient types. Congruence ..."
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. We introduce a typetheoretical framework in which canonical term rewriting systems can be represented faithfully both from the logical and the computational points of view. The framework is based on congruence types, a new syntax which combines inductive, algebraic and quotient types. Congruence types improve on existing work to combine type theories with algebraic rewriting by making explicit the fact that the termrewriting systems under consideration are initial models of an equational theory. As a result, the interaction gustavo:thesisween the type theory and the algebraic types (rewriting systems) is much more powerful than in previous work. Congruence types can be used (i) to introduce initial models of canonical termrewriting systems (ii) to obtain a suitable computational behavior of a definable operation (iii) to provide an elegant solution to the problem of equational reasoning in type theory. 1 Introduction The combination of type systems with algebraic rewriting system...
Formalizing type operations using the “Image” type constructor
 Workshop on Logic, Language, Information and Computation (WoLLIC
, 2006
"... In this paper we introduce a new approach to formalizing certain type operations in type theory. Traditionally, many type constructors in type theory are independently axiomatized and the correctness of these axioms is argued semantically. In this paper we introduce a notion of an “image ” of a give ..."
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In this paper we introduce a new approach to formalizing certain type operations in type theory. Traditionally, many type constructors in type theory are independently axiomatized and the correctness of these axioms is argued semantically. In this paper we introduce a notion of an “image ” of a given type under a mapping that captures the spirit of many of such semantical arguments. This allows us to use the new “image ” type to formalize within the type theory a large range of type constructors that were traditionally formalized via postulated axioms. We demonstrate the ability of the “image ” constructor to express “forgetful ” types by using it to formalize the “squash ” and “set ” type constructors. We also demonstrate its ability to handle types with nontrivial equality relations by using it to formalize the union type operator. We demonstrate the ability of the “image ” constructor to express certain inductive types by showing how the type of lists and a higherorder abstract syntax type can be naturally formalized using the new type constructor. The work presented in this paper have been implemented in the MetaPRL proof assistant and all the derivations checked by MetaPRL.
OpenEndedness of Objects and Types in MartinL\"of’s Type Theory
"... This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formu ..."
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This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formulation of computational openendedness. Suppose that a language underlying the theory is specified as a method system, which consists of a preobject system as the computational part and a pretype system as the structural part. Then types and their objects are unifoImly and inductively constructed as a type system that is built from the method system and that can provide a semantics of the theory. The main theorem shows that the original inference rules concerning objects or types remain valid in any type system built from a deterministic and regular extension of the original method system. This result includes a prescription for the class of types that can be introduced into the theory, which prescription is useful for checking whether specific new types can be introduced. 1