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Little Theories
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable wa ..."
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In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
A HOL theory of Euclidean space
 In Hurd and Melham [7
, 2005
"... Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in ..."
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Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in a simple and useful way, even though HOL does not permit dependent types. In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, e.g. checking compatibility in matrix multiplication. Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary Ndimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions. 1 1 The problem with R N
Elements of Mathematical Analysis in PVS
 Ninth international Conference on Theorem Proving in Higher Order Logics TPHOL
, 1996
"... . This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS i ..."
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. This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS including recent extensions of the type system and discusses their merits and effectiveness. We conclude by a brief comparison with similar developments using other theorem provers. 1 Introduction PVS is a specification and verification system whose ambition is to make formal proofs practical and applicable to large and complex problems. The system is based on a variant of higher order logic which includes complex typing mechanisms such as predicate subtypes or dependent types. It offers an expressive specification language coupled with a theorem prover designed for efficient interactive proof construction. In previous work we have applied PVS to the requirements analysis of a substantially ...
CORE and HULL Constructors in Gödel’s Class Theory
"... Abstract. The GOEDEL program, a computer implementation of Gödel’s algorithm for class formation in Mathematica TM was used for formulating definitions and discovering theorems about topology and its generalizations, working within Gödel’s class theory. A general characterization of CORE[x] and HULL ..."
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Abstract. The GOEDEL program, a computer implementation of Gödel’s algorithm for class formation in Mathematica TM was used for formulating definitions and discovering theorems about topology and its generalizations, working within Gödel’s class theory. A general characterization of CORE[x] and HULL[x] functions discovered in the course of this work is the primary focus of this paper. 1
unknown title
, 1992
"... Abstract In the "little theories " version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach ..."
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Abstract In the &quot;little theories &quot; version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
A Simple Virtual Memory Scheme Formalized in IMPS
"... . In this paper we formalize a simple virtual memory scheme derived from Mach and Multics. It is represented by state machine operations in imps, an Interactive Mathematical Proof System. We prove that a store with a global page table faithfully renes an abstract machine in which processes may p ..."
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. In this paper we formalize a simple virtual memory scheme derived from Mach and Multics. It is represented by state machine operations in imps, an Interactive Mathematical Proof System. We prove that a store with a global page table faithfully renes an abstract machine in which processes may perform fetch and store operations against a set of persistent memory objects. Both safety conditions and liveness conditions are proved. Some ne points treated in the model include: the niteness of virtual address spaces and physical store; the ability to map a portion of a permanent memory object into the address space; initialization of newly allocated memory to zero; and the need for pagealigned addresses in some operations. The paper has two main purposes. First, it illustrates how naturally imps can model this sort of problem, can prove the necessary theorems, and can present the results. Technical details are presented as typeset automatically by imps. Second, it illustra...