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The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equatio ..."
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
An executable formalization of the HOL/Nuprl connection in the metalogical framework Twelf
 In Geoff Sutcliffe and Andrei Voronkov, editors, Proceedings of Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), Montego
, 2005
"... Abstract. Howe’s HOL/Nuprl connection is an interesting example of a translation between two fundamentally different logics, namely a typed higherorder logic and a polymorphic extensional type theory. In earlier work we have established a prooftheoretic correctness result of the translation in a w ..."
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Abstract. Howe’s HOL/Nuprl connection is an interesting example of a translation between two fundamentally different logics, namely a typed higherorder logic and a polymorphic extensional type theory. In earlier work we have established a prooftheoretic correctness result of the translation in a way that complements Howe’s semanticsbased justification and furthermore goes beyond the original HOL/Nuprl connection by providing the foundation for a proof translator. Using the Twelf logical framework, the present paper goes one step further. It presents the first rigorous formalization of this treatment in a logical framework, and hence provides a safe alternative to the translation of proofs. 1
An Executable Formalization of the HOL/Nuprl Connection
 in the Metalogical Framework Twelf. LPAR 2004
"... Abstract. Howe’s HOL/Nuprl connection is an interesting example of a translation between two fundamentally different logics, namely a typed higherorder logic and a polymorphic extensional type theory. In earlier work we have established a prooftheoretic correctness result of the translation in a w ..."
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Abstract. Howe’s HOL/Nuprl connection is an interesting example of a translation between two fundamentally different logics, namely a typed higherorder logic and a polymorphic extensional type theory. In earlier work we have established a prooftheoretic correctness result of the translation in a way that complements Howe’s semanticsbased justification and furthermore goes beyond the original HOL/Nuprl connection by providing the foundation for a proof translator. Using the Twelf logical framework, the present paper goes one step further. It presents the first rigorous formalization of this treatment in a logical framework, and hence provides a safe alternative to the translation of proofs. 1
Abstract Innovations in Computational Type Theory using
"... For twenty years the Nuprl (“new pearl”) system has been used to develop software systems and formal theories of computational mathematics. It has also been used to explore and implement computational type theory (CTT) – a formal theory of computation closely related to MartinLöf’s intuitionistic ..."
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For twenty years the Nuprl (“new pearl”) system has been used to develop software systems and formal theories of computational mathematics. It has also been used to explore and implement computational type theory (CTT) – a formal theory of computation closely related to MartinLöf’s intuitionistic type theory (ITT) and to the calculus of inductive constructions (CIC) implemented in the Coq prover. This article focuses on the theory and practice underpinning our use of Nuprl for much of the last decade. We discuss innovative elements of type theory, including new type constructors such as unions and dependent intersections, our theory of classes, and our theory of event structures. We also discuss the innovative architecture of Nuprl as a distributed system and as a transactional database of formal mathematics using the notion of abstract object identifiers. The database has led to an independent project called the Formal Digital Library, FDL, now used as a repository for Nuprl results as well as selected results from HOL, MetaPRL, and PVS. We discuss Howe’s set theoretic semantics that is used to relate such disparate theories and systems as those represented by these provers. 1
NummSquared: a New Foundation for Formal Methods
, 2006
"... To spread the use of formal methods, a language must appeal to programmers, mathematicians and logicians. Set theory is the standard mathematical foundation, but often ignores computational aspects. Type theory has good support for mixing specification and implementation, but often imposes type cons ..."
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To spread the use of formal methods, a language must appeal to programmers, mathematicians and logicians. Set theory is the standard mathematical foundation, but often ignores computational aspects. Type theory has good support for mixing specification and implementation, but often imposes type constraints in excess of those found in typical programming languages. Furthermore, standard mathematics is untyped. Languages based on the untyped lambda calculus often permit nonterminating programs and require reasoning in nonclassical logics. Languages without higher order functions often lack polymorphism. There is a wide gap between conventional programming languages and logic. This paper proposes NummSquared, a new formal language based only on untyped higher order functions, which allows only terminating programs, has a classical logic, is related to wellfounded set theory, and supports reflection. NummSquared supports rapid prototyping without proofs to reduce cost, and supports adding proofs later. 1 Overview and comparison A feature that is elegant in a programming language may be inappropriate for formal methods, particularly when the programming language must also serve as a logic. The untyped lambda calculus, a useful model for many programming languages, is elegant because it is based only on untyped functions, and any function may be passed as an ar