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A Comparative Study of Coq and HOL
 In Gunter and Felty [GF97
, 1997
"... . This paper illustrates the differences between the style of theory mechanisation of Coq and of HOL. This comparative study is based on the mechanisation of fragments of the theory of computation in these systems. Examples from these implementations are given to support some of the arguments discus ..."
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. This paper illustrates the differences between the style of theory mechanisation of Coq and of HOL. This comparative study is based on the mechanisation of fragments of the theory of computation in these systems. Examples from these implementations are given to support some of the arguments discussed in this paper. The mechanisms for specifying definitions and for theorem proving are discussed separately, building in parallel two pictures of the different approaches of mechanisation given by these systems. 1 Introduction This paper compares the different theorem proving approaches of the HOL [10] and Coq [5] proof assistants. This comparison is based on a case study involving the mechanisation of parts of the theory of computation in the two systems. This paper does not illustrate these mechanisations but rather discusses the differences between the two systems and backs up certain points by examples taken from the case studies. One motivation of this work is that many users of theo...
The HOLOmega Logic
"... Abstract. A new logic is posited for the widely used HOL theorem prover, as an extension of the existing higher order logic of the HOL4 system. The logic is extended to three levels, adding kinds to the existing levels of types and terms. New types include type operator variables and universal types ..."
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Abstract. A new logic is posited for the widely used HOL theorem prover, as an extension of the existing higher order logic of the HOL4 system. The logic is extended to three levels, adding kinds to the existing levels of types and terms. New types include type operator variables and universal types as in System F. Impredicativity is avoided through the stratification of types by ranks according to the depth of universal types. The new system, called HOLOmega or HOLω, isamergingofHOL4, HOL2P[11], and major aspects of System Fω from chapter 30 of [10]. This document presents the abstract syntax and semantics for the kinds, types, and terms of the logic, as well as the new fundamental axioms and rules of inference. As the new logic is constructed according to the design principles of the LCF approach, the soundness of the entire system depends critically and solely on the soundness of this core. 1
HOL2P  A system of classical higher order logic with second order polymorphism
 20th International Conference on Theorem Proving in Higher Order Logics: TPHOLs 2007, volume 4732 of Lecture Notes in Computer Science
, 2007
"... Abstract. This paper introduces the logical system HOL2P that extends classical higher order logic (HOL) with type operator variables and universal types. HOL2P has explicit term operations for type abstraction and type application. The formation of type application terms t [ T] is restricted to sma ..."
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Abstract. This paper introduces the logical system HOL2P that extends classical higher order logic (HOL) with type operator variables and universal types. HOL2P has explicit term operations for type abstraction and type application. The formation of type application terms t [ T] is restricted to small types T that do not contain any universal types. This constraint ensures the existence of a settheoretic model and thus consistency. The expressiveness of HOL2P allows categorytheoretic concepts such as natural transformations and initial algebras to be applied at the level of polymorphic HOL functions. The parameterisation of terms with type operators adds genericity to theorems. Type variable quantification can also be expressed. A prototype of HOL2P has been implemented on top of HOLLight. Type inference is semiautomatic, and some type annotations are necessary. Reasoning is supported by appropriate tactics. The implementation has been used to check some sample derivations. 1
A Reference Version of HOL
"... . The second author has implemented a reference version of the HOL logic (henceforth called gtt). This version, written in Standard ML, is as simple as possible, making as few assumptions as necessary to present the essence of HOL. This simplicity makes the implementation easy to understand, to port ..."
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. The second author has implemented a reference version of the HOL logic (henceforth called gtt). This version, written in Standard ML, is as simple as possible, making as few assumptions as necessary to present the essence of HOL. This simplicity makes the implementation easy to understand, to port, to develop, to change, and to informally reason about. The first author has ported gtt to another dialect of ML, and developed the parsing, prettyprinting, and typechecking support needed to take gtt beyond its initial rudimentary conception. The implementation of gtt has already been of use in developing a variant of the HOL logic. As of this writing, there are at least four or five extant implementations of the HOL logic. These have been intensively developed, in some cases over decades, which leads us to an overwhelming question: why another? In particular, why gtt? There are several answers to this, stemming from different desires and needs in the HOL community. Changing the logic a ...
Toward a Super Duper Hardware Tactic
, 1993
"... We present techniques for automating many of the tedious aspects of hardware verification in a higher order logic theorem proving environment. We employ two complementary approaches. The first involves intelligent tactics which incorporate many of the smaller steps currently applied by the user. ..."
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We present techniques for automating many of the tedious aspects of hardware verification in a higher order logic theorem proving environment. We employ two complementary approaches. The first involves intelligent tactics which incorporate many of the smaller steps currently applied by the user. The second uses hardware combinators to partially automate inductive proofs for iterated hardware structures. We envision a system that captures most of this reasoning in one tactic, SuperDuperHWTac. Ideally, users would use this tactic on a goal for proving that a hardware component meets its specification, and get back a proof documented at a level they would have written by hand. This paper presents preliminary work toward SuperDuperHWTac in both the HOL and Nuprl proof development systems. 1 Introduction Higher order logic makes specifying hardware designs natural. Unfortunately, it also makes verification tedious. If verification engineers adopt a specific style for doing hardwa...
HOL Done Right
, 1995
"... In our opinion, history and compatibility considerations have rendered existing HOL implementations rather messy and badly organized. We describe how, building on joint work with Konrad Slind, we have produced a reengineered HOL. Various experiments have been tried on this ‘toy ’ version, and we wi ..."
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In our opinion, history and compatibility considerations have rendered existing HOL implementations rather messy and badly organized. We describe how, building on joint work with Konrad Slind, we have produced a reengineered HOL. Various experiments have been tried on this ‘toy ’ version, and we will report the results. 1