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On Formalised Proofs of Termination of Recursive Functions
 In Proceedings of the Int. Conf. on Principles and Practice of Declarative Programming, volume 1702 of LNCS
, 1999
"... In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive de nitions of functions are shown to terminate automatically. ..."
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In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive de nitions of functions are shown to terminate automatically.
On automating process algebra proofs
 Proceedings of the 11th International Symposium on Computer and Information Sciences, ISCIS XI
, 1996
"... In [10] Groote and Springintveld incorporated several modeloriented techniques { such asinvariants, matching criteria, state mappings { in the processalgebraic framework of CRL for structuring and simplifying protocol veri cations. In this paper, we formalise these extensions in Coq, which is a pr ..."
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In [10] Groote and Springintveld incorporated several modeloriented techniques { such asinvariants, matching criteria, state mappings { in the processalgebraic framework of CRL for structuring and simplifying protocol veri cations. In this paper, we formalise these extensions in Coq, which is a proof development tool based on type theory. In the updated framework, the length of proof constructions is reduced significantly. Moreover, the new approach allows for more automation (proof generation) than was possible in the past. The results are illustrated by an example in which we prove two queue representations equal. 1
Adding the axioms to Axiom: Towards a system of automated reasoning in Aldor
 Computing Laboratory, University of Kent
, 1998
"... A number of combinations of theorem proving and computer algebra systems have been proposed; in this paper we describe another, namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor { the Axiom Library Compiler { and show that with some modica ..."
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A number of combinations of theorem proving and computer algebra systems have been proposed; in this paper we describe another, namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor { the Axiom Library Compiler { and show that with some modications we can use the dependent types of the system to model a logic, under the CurryHoward isomorphism. We give a number of example applications of the logic we construct. 1
Subtyping and Inheritance for Inductive Types
 In Informal proceedings of the 1994 TYPES Workshop
, 1997
"... Inheritance and subtyping are key features of objectoriented languages. We show that there are corresponding  or, more precisely, dual  notions for inductive types (or algebraic datatypes): there is a natural notion of subtyping for these types and an associated form of code reuse (inheritance) ..."
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Inheritance and subtyping are key features of objectoriented languages. We show that there are corresponding  or, more precisely, dual  notions for inductive types (or algebraic datatypes): there is a natural notion of subtyping for these types and an associated form of code reuse (inheritance) for programs on these types. Inheritance and subtyping for inductive types not only suggest possible extensions of functional programming languages, but also provide a new perspective on inheritance as we know it from objectoriented languages, which may help to get a better understanding of this notion. 1 Introduction Functional programming languages such as ML [MTH90] provide algebraic datatypes  e.g. lists, trees  , and type theories such as Coq [Cor95] or Alf [AGNvS94] provide a more general notion of inductive type. Algebras are a wellknown way of modelling these types: algebraic datatypes and inductive types can be understood as term algebras (or initial algebras). It has been o...
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...
On automating inductive and noninductive termination methods
 In Proceedings of the 5th Asian Computing Science Conference, volume 1742 of LNCS
, 1999
"... Abstract. The Coq and ProPre systems show the automated termination of a recursive function by rst constructing a tree associated with the specication of the function which satises a notion of terminal property and then verifying that this construction process is formally correct. However, those t ..."
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Abstract. The Coq and ProPre systems show the automated termination of a recursive function by rst constructing a tree associated with the specication of the function which satises a notion of terminal property and then verifying that this construction process is formally correct. However, those two steps strongly depend on inductive principles and hence Coq and ProPre can only deal with the termination proofs that are inductive. There are however many functions for which the termination proofs are noninductive. In this article, we attempt to extend the class of functions whose proofs can be done automatically a la Coq and ProPre to a larger class including functions whose termination proofs are not inductive. We do this by extending the terminal property notion and replacing the verication step above by one that searches for a decreasing measure which can be used to establish the termination of the function. 1
A Mechanisation of Computability Theory in HOL
 In Proceedings of the 9th International Conference on Theorem Proving in Higher Order Logics
, 1996
"... . This paper describes a mechanisation of computability theory in HOL using the Unlimited Register Machine (URM) model of computation. The URM model is first specified as a rudimentary machine language and then the notion of a computable function is derived. This is followed by an illustration o ..."
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. This paper describes a mechanisation of computability theory in HOL using the Unlimited Register Machine (URM) model of computation. The URM model is first specified as a rudimentary machine language and then the notion of a computable function is derived. This is followed by an illustration of the proof of a number of basic results of computability which include various closure properties of computable functions. These are used in the implementation of a mechanism which partly automates the proof of the computability of functions and a number of functions are then proved to be computable. This work forms part of a comparative study of different theorem proving approaches and a brief discussion regarding theorem proving in HOL follows the description of the mechanisation. 1 Introduction The theory of computation is a field which has been widely explored in mathematical and computer science literature [4, 12, 13] and several approaches to a standard model of computation h...
Integrating Computer Algebra and Reasoning through the Type System of Aldor
, 2000
"... . A number of combinations of reasoning and computer algebra systems have been proposed; in this paper we describe another, namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor { the Axiom Library Compiler { and show that with some modicat ..."
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. A number of combinations of reasoning and computer algebra systems have been proposed; in this paper we describe another, namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor { the Axiom Library Compiler { and show that with some modications we can use the dependent types of the system to model a logic, under the CurryHoward isomorphism. We give a number of example applications of the logic we construct and explain a prototype implementation of a modied typechecking system written in Haskell. 1 Introduction Symbolic mathematical { or computer algebra { systems, such as Axiom [13], Maple and Mathematica, are in everyday use by scientists, engineers and indeed mathematicians, because they provide a user with techniques of, say, integration which far exceed those of the person themselves, and make routine many calculations which would have been impossible some years ago. These systems are, moreover, taught as standar...
A proof of the S m n theorem in Coq
, 1997
"... This report describes the implementation of a mechanisation of the theory of computation in the Coq proof assistant which leads to a proof of the S m n theorem. This mechanisation is based on a model of computation similar to the partial recursive function model and includes the denition of a comput ..."
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This report describes the implementation of a mechanisation of the theory of computation in the Coq proof assistant which leads to a proof of the S m n theorem. This mechanisation is based on a model of computation similar to the partial recursive function model and includes the denition of a computable function, proofs of the computability of a number of functions and the denition of an eective coding from the set of partial recursive functions to natural numbers. This work forms part of a comparative study of the HOL and Coq proof assistants.
A Computer Checked Algebraic Verification of a Distributed Summation Algorithm
"... Reports are available at: ..."