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Translating Dependent Type Theory into Higher Order Logic
 IN PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON TYPED LAMBDA CALCULI AND APPLICATIONS, VOLUME 664 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... This paper describes a translation of the complex calculus of dependent type theory into the relatively simpler higher order logic originally introduced by Church. In particular, it shows how type dependency as found in MartinLöf's Intuitionistic Type Theory can be simulated in the formulat ..."
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This paper describes a translation of the complex calculus of dependent type theory into the relatively simpler higher order logic originally introduced by Church. In particular, it shows how type dependency as found in MartinLöf's Intuitionistic Type Theory can be simulated in the formulation of higher order logic mechanized by the HOL theoremproving system. The outcome is a theorem prover for dependent type theory, built on top of HOL, that allows natural and flexible use of settheoretic notions. A bit more technically, the language of the resulting theoremprover is the internal language of a (boolean) topos (as formulated by Phoa).
Automatic Proof and Disproof in Isabelle/HOL
, 2011
"... Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the c ..."
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Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the counterexample generator Quickcheck uses the ML compiler as a fast evaluator for ground formulas, and its rival Nitpick is based on the model finder Kodkod, which performs a reduction to SAT. Together with the Isar structured proof format and a new asynchronous user interface, these tools have radically transformed the Isabelle user experience. This paper provides an overview of the main automatic proof and disproof tools.
Formal verification of square root algorithms
 Formal Methods in Systems Design
, 2003
"... Abstract. We discuss the formal verification of some lowlevel mathematical software for the Intel ® Itanium ® architecture. A number of important algorithms have been proven correct using the HOL Light theorem prover. After briefly surveying some of our formal verification work, we discuss in more ..."
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Abstract. We discuss the formal verification of some lowlevel mathematical software for the Intel ® Itanium ® architecture. A number of important algorithms have been proven correct using the HOL Light theorem prover. After briefly surveying some of our formal verification work, we discuss in more detail the verification of a square root algorithm, which helps to illustrate why some features of HOL Light, in particular programmability, make it especially suitable for these applications. 1. Overview The Intel ® Itanium ® architecture is a new 64bit architecture jointly developed by Intel and HewlettPackard, implemented in the Itanium® processor family (IPF). Among the software supplied by Intel to support IPF processors are some optimized mathematical functions to supplement or replace less efficient generic libraries. Naturally, the correctness of the algorithms used in such software is always a major concern. This is particularly so for division, square root and certain transcendental function kernels, which are intimately tied to the basic architecture. First, in IA32 compatibility mode, these algorithms are used by hardware instructions like fptan and fdiv. And while in “native ” mode, division and square root are implemented in software, typical users are likely to see them as part of the basic architecture. The formal verification of some of the division algorithms is described by Harrison (2000b), and a representative verification of a transcendental function by Harrison (2000a). In this paper we complete the picture by considering a square root algorithm. Division, transcendental functions and square roots all have quite distinctive features and their formal verifications differ widely from each other. The present proofs have a number of interesting features, and show how important some theorem prover features — in particular programmability — are. The formal verifications are conducted using the freely available 1 HOL Light prover (Harrison, 1996). HOL Light is a version of HOL (Gordon and Melham, 1993), itself a descendent of Edinburgh LCF
HOL Light Tutorial (for version 2.20)
, 2007
"... The HOL Light theorem prover can be difficult to get started with. While the manual is fairly detailed and comprehensive, the large amount of background information that has to be absorbed before the user can do anything interesting is intimidating. Here we give an alternative ‘quick start ’ guide, ..."
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The HOL Light theorem prover can be difficult to get started with. While the manual is fairly detailed and comprehensive, the large amount of background information that has to be absorbed before the user can do anything interesting is intimidating. Here we give an alternative ‘quick start ’ guide, aimed at teaching basic use of the system quickly by means of a graded set of examples. Some readers may find it easier to absorb; those who do not are referred after all to the standard manual. “Shouldn’t we read the instructions?”
A HOL specification of the ARM instruction set architecture
, 2001
"... This report gives details of a hol specification of the arm instruction set architecture. It is shown that the hol proof tool provides a suitable environment in which to model the architecture. The specification is used to execute fragments of arm code generated by an assembler. The specification is ..."
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This report gives details of a hol specification of the arm instruction set architecture. It is shown that the hol proof tool provides a suitable environment in which to model the architecture. The specification is used to execute fragments of arm code generated by an assembler. The specification is based primarily around the third version of the arm architecture, and the intent is to provide a target semantics for future microprocessor verifications. Contents 1
Refinement Concepts Formalized in Higher Order Logic
 Formal Aspects of Computing
, 1989
"... A theory of commands with weakest precondition semantics is formalized using the HOL proof assistant system. The concept of refinement between commands is formalized, a number of refinement rules are proved and it is shown how the formalization can be used for proving refinements of actual program t ..."
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A theory of commands with weakest precondition semantics is formalized using the HOL proof assistant system. The concept of refinement between commands is formalized, a number of refinement rules are proved and it is shown how the formalization can be used for proving refinements of actual program texts correct. 1 Introduction The refinement calculus is a theory of program transformations that preserve the total correctness of programs. It was first described in [Ba78, Ba80] and has been further elaborated in [Ba88a, Ba88b, MoRoGa88, Morr87]. It is based on the weakest precondition technique of [Di76]. The refinement calculus has been used as a tool for stepwise refinement of sequential algorithms, and recently also for the derivation of parallel algorithms from sequential algorithms [BaSe89, Wr89]. The HOL system (Higher Order Logic) is a theorem proving assistant, which can be used to formalize theories and verify proofs of theorems within these theories. It is based on the LCF syst...
A Mechanized Theory of the picalculus in HOL
, 1992
"... : The ßcalculus is a process algebra for modelling concurrent systems in which the pattern of communication between processes may change over time. This paper describes the results of preliminary work on a definitional formal theory of the ßcalculus in higher order logic using the HOL theorem prov ..."
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: The ßcalculus is a process algebra for modelling concurrent systems in which the pattern of communication between processes may change over time. This paper describes the results of preliminary work on a definitional formal theory of the ßcalculus in higher order logic using the HOL theorem prover. The ultimate goal of this work is to provide practical mechanized support for reasoning with the ßcalculus about applications. Introduction The ßcalculus [17, 18] is a process algebra proposed by Milner, Parrow and Walker for modelling concurrent systems in which the pattern of interconnection between processes may change over time. This paper describes work on a mechanized formal theory of the ßcalculus in higher order logic using the HOL theorem prover [8]. The main aim of this work is to construct a practical and sound theoremproving tool to support reasoning about applications using the ßcalculus, as well as metatheoretic reasoning about the ßcalculus itself. Four general prin...
Proof Synthesis and Reflection for Linear Arithmetic
 J. OF AUT. REASONING
"... This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in t ..."
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This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proofproducing functional program, and once by reflection, i.e. by computations inside the logic rather than in the metalanguage. Both formalizations are generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster.
A Formal Approach to Component Adaptation and Composition
, 2005
"... Component based software engineering (CBSE), can in principle lead to savings in the time and cost of software development, by encouraging software reuse. However the reality is that CBSE has not been widely adopted. From a technical perspective, the reason is largely due to the di#culty of locating ..."
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Component based software engineering (CBSE), can in principle lead to savings in the time and cost of software development, by encouraging software reuse. However the reality is that CBSE has not been widely adopted. From a technical perspective, the reason is largely due to the di#culty of locating suitable components in the library and adapting these components to meet the specific needs of the user.