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Equality In Lazy Computation Systems
, 1998
"... In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equ ..."
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Cited by 95 (6 self)
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In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equality reasoning are valid for it. This condition is a simple syntactic one, and is easy to verify for the various lazy computation systems we have considered so far. We also give conditions under which the equivalence coincides with observational congruence. These results have some important consequences for type theories like those of MartinLöf and Nuprl.
Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra
, 1995
"... This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of rou ..."
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Cited by 44 (4 self)
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This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of roughly 500 definitions and 1300 theorems. Much of this material is of a foundational nature and supports all current work in Nuprl. This thesis concentrates on describing the half of this corpus that is concerned with abstract algebra and that covers topics central to the mathematics of the co...
Reasoning About Functional Programs in Nuprl
 In Functional Programming, Concurrency, Simulation and Automated Reasoning
, 1993
"... . There are two ways of reasoning about functional programs in the constructive type theory of the Nuprl proof development system. Nuprl can be used in a conventional programverification mode, in which functional programs are written in a familiar style and then proven to be correct. It can als ..."
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Cited by 12 (0 self)
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. There are two ways of reasoning about functional programs in the constructive type theory of the Nuprl proof development system. Nuprl can be used in a conventional programverification mode, in which functional programs are written in a familiar style and then proven to be correct. It can also be used in an extraction mode, where programs are not written explicitly, but instead are extracted from mathematical proofs. Nuprl is the only constructive type theory to support both of these approaches. These approaches are illustrated by applying Nuprl to Boyer and Moore's "majority" algorithm. 1 Introduction A type system for a functional programming language can be syntactic or semantic. In a syntactically typed language, such as SML 1 [25], typing is a property of the syntax of expressions. Only certain combinations of language constructs are designated "welltyped", and only welltyped expressions are given a meaning. Each welltyped expression has a type which can be derive...
Partial computations in constructive type theory
 JOURNAL OF LOGIC AND COMPUTATION
, 1991
"... Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects ..."
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Cited by 7 (5 self)
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Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A fixed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of fixed points. There is a Scott fixed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.
Importing Isabelle Formal Mathematics into NuPRL
, 1999
"... 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. ..."
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Cited by 6 (0 self)
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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.
Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...
Some Normalization Properties of MartinLof's Type Theory, and Applications
 in Proc. 1st Internat. Conf. on Theoretical Aspects of Computer Software, Lecture Notes in Computer Science
, 1991
"... For certain kinds of applications of type theories, the faithfulness of formalization in the theory depends on intensional, or structural, properties of objects constructed in the theory. For type theories such as LF, such properties can be established via an analysis of normal forms and types. In t ..."
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Cited by 3 (1 self)
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For certain kinds of applications of type theories, the faithfulness of formalization in the theory depends on intensional, or structural, properties of objects constructed in the theory. For type theories such as LF, such properties can be established via an analysis of normal forms and types. In type theories such as Nuprl or MartinLof's polymorphic type theory, which are much more expressive than LF, the underlying programming language is essentially untyped, and terms proved to be in types do not necessarily have normal forms. Nevertheless, it is possible to show that for MartinLof's type theory, and a large class of extensions of it, a sufficient kind of normalization property does in fact hold in certain wellbehaved subtheories. Applications of our results include the use of the type theory as a logical framework in the manner of LF, and an extension of the proofsasprograms paradigm to the synthesis of verified computer hardware. For the latter application we point out some ...
Nuprl and its Use in Circuit Design
, 1992
"... Nuprl is an interactive theorem proving system in the LCF tradition. It has a higher order logic and a very expressive type theory; the type theory includes dependent function types (\Pi types), dependent product types (\Sigma types) and set types. Nuprl also has a well developed XWindows user inte ..."
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Cited by 2 (0 self)
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Nuprl is an interactive theorem proving system in the LCF tradition. It has a higher order logic and a very expressive type theory; the type theory includes dependent function types (\Pi types), dependent product types (\Sigma types) and set types. Nuprl also has a well developed XWindows user interface and allows for the use of clear and concise notations, close to ones used in print. Proofs are objects which can be viewed, and serve as readable explanations of theorems. Tactics provide a highlevel extendible toolkit for proof development, while the soundness of the system relies only a fixed set of rules. We give an overview of the Nuprl system, focusing in particular on the advantages that the type theory brings to formal methods for circuit design. We also discuss ongoing projects in verifying floatingpoint circuits, verifying the correctness of hardware synthesis systems, and synthesizing circuits by exploiting the constructivity of Nuprl's logic. Keyword Codes: F.4.1; B.6.2; I...
Formalization of Divisibility Theory in Nuprl
"... The formalization of divisibility theory over cancellation monoids in Nuprl is described. The main theorems presented concern the existence and uniqueness of factorisations. Issues addressed include how to make formalized mathematics readable and the use of automated inference. The constructive nat ..."
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The formalization of divisibility theory over cancellation monoids in Nuprl is described. The main theorems presented concern the existence and uniqueness of factorisations. Issues addressed include how to make formalized mathematics readable and the use of automated inference. The constructive nature of mathematics in Nuprl is also discussed.