J. Harrison, L. Th'ery. Extending HOL Theorem Prover with a Computer Algebra System to Reason About the Reals. In Proceedings of Higher Order Logic Theorem Proving and its Applications. Editors: J.J. Joyce, C. Seger Lecture Notes in Computer Science 780, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

...., KoMeT, Otter, Setheo and Spass) which are called depending on the problem at hand. PVS [OSR93] invokes an external BDD package for Boolean simplification and a model checker for CTL formulas. There are only few examples of a cooperation in which results are effectively verified: Harrison [HT93] describes an interface between the HOL theorem prover and a computer algebra system, in which the result returned by the computer algebra system, such as a factorization of a polynomial, is checked for validity. Recently, there are efforts to integrate a proof search procedure for intuitionistic ....

....t i =T t i 1 can be derived. Essentially, the transitivity of equality is used to concatenate the proof steps. trJust( Gamma ; AxiomMap; t 0 0 = t 0 1 ; just) Gamma p 0 : t 0 =T t 1 trSubProof ( Gamma ; AxiomMap; splist) Gamma p 1 : t 1 =T t 2 trSubProof ( Gamma ; AxiomMap; ht 0 0 = t 0 1 ; justi : splist) Gamma (equal transitive T t 0 t 1 t 2 p 0 p 1 ) t 0 =T t 2 For example, when translating Lemma 1 in Figure 3, the equality y = op(unit( y) is translated first, then (by a repeated application of the transitivity rule) the equality op(unit( y) ....

John Harrison and Laurent Th'ery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In Jeffrey J. Joyce and Carl Seger, editors, Proceedings of the

.... Two Level Approach towards Lean Proof Checking Gilles Barthe # , Mark Ruys and Henk Barendregt May 15, 1996 Abstract We present a simple and e#ective methodology for equational reasoning in proof checkers. The method is based on a two level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with ....

....test. This method is called the autarkic way because it does not involve any change to the type theory or the proof checker. It must be said that this method seems currently too ine#cient to be used in practice. Most proposals in the literature opt for the external believing approach [2, 15, 17] Indeed, the external believing way has an obvious advantage: hybrid systems o#er a shortcut to integrate term rewriting in proof checking. However, the approach has two disadvantages: proof checkers are based on well understood languages whose logical and computational status are well ....

[Article contains additional citation context not shown here]

J. Harrison and L. Thery. Extending the HOL theorem prover with a computer algebra system to reason about the reals, in proceedings of HOL'93, LNCS, 1993.

....to strengthen interactive proving methods (and tools) by adding automatic methods (and tools) which are able to handle a relevant amount of subtasks. Apart from rst order logic theorem proving, the most important reasoning methods being integrated in interactive frameworks are: computer algebra [24, 2], model checking [27, 30] and decision procedures [11] Each of them is typically applied to some particular domain(s) Computer algebra is incorporated into proof systems solving problems of mathematical or physical origin. These applications are very di erent from the problems occurring in ....

J. Harrison and L. Thery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In J. J. Joyce and C.-J. H. Seger, editors, Higher Order Logic Theorem Proving and its Applications: 6th International Workshop, HUG'93, Vancouver, Canada, LNCS 780, pages 174-184. Springer-Verlag, 1993.

....same effect by creating and additional datatype of lazy thm. This was a method external to the logic and required a fair amount of duplication of functions for theorems as functions for lazy theorems. John Harrison went on to use this method in his work coupling HOL with computer algebra systems [3]. The advantage of Richard Boulton s work is that it requires no change to the logic. Our work does require a change to the logic, but is it provably consistent and we feel is actually much more light weight. It also provides many of the advantages of his system; implementing John Harrison s work ....

J. Harrison and L. Th'ery. Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals. In Higher Order Logic Theorem Proving and Its Applications. Springer-Verlag, 1993.

....progress to define translations of certain HOL goals to input for a resolution prover. Joyce and Seger [43] describe a link to the Voss model checking tool to be used in hardware verification. Rajan, Shankar and Srivas [64] describe the integration of a model checker with PVS. Harrison and Th ery [39], and Ballarin, Homann and Calmet [6] report experiments of integration of the computer algebra system Maple [15] with HOL and Isabelle [62] respectively. Analytica [18] is a theorem prover, written in the Mathematica [66] environment, which uses the powerful algorithms embedded in Mathematica to ....

.... including: the Nelson Oppen cooperating decision procedures and simplifier; resolution [16] and Alan Bundy s rippling [14] Another important area of application is the integration of computer algebra systems and theorem provers, an area which has lately received a lot of interest (see e.g. [18, 6, 39, 70]) A first attempt to apply our approach to this problem domain is reported in [40] Some preliminary work of ours in this direction is reported in [31] In this area the work on the definition of the interaction level will exploit the results described in [57] and also the results of the OpenMath ....

J. Harrison and L. Th'ery. Extending the HOL Theorem Prover with a Computer Algebra System to Reason about the Reals. In Joyce and Seger [42], pages 174--184.

....and propositional temporal formulas, and a semi decision procedure for first order logic. The experience of integrating a linear arithmetic module into NQTHM is described in detail in [17] Rajan, Shankar and Srivas [169] describe the integration of a model checker with PVS. Harrison and Th ery [72], and Ballarin, Homann and Calmet [7] report experiments of integration of the computer algebra system Maple [21] with HOL and Isabelle [166] respectively. These experiments clearly demonstrate the need for composite, heterogeneous systems and the ability to both tightly integrate and loosely ....

J. Harrison and L. Th'ery. Extending the HOL Theorem Prover with a Computer Algebra System to Reason about the Reals. In Claesen and Gordon [27], pages 174--184. 29

....is often graduate level work . Notice too that theorem provers prove theorems rather than make conjectures or do calculations, and a proof of a simple arithmetical statement, even if performed entirely automatically by means of a suitable tactic, may still be lengthy. Harrison and Th ery [27] experimented with combining such development with the use of Maple as an oracle to compute, for example, factorisations of polynomials or inde nite integrals which could then by veri ed by the prover. However the success of theorem proving in applications like the veri cation of hardware has ....

....a great loss in eciency if standard algorithms such as factorisation are implemented from rst principles. Various hybrid systems have been proposed, where theorem provers call computer algebra systems [28] either using the CAS as an oracle whose results are then then proved in the theorem prover[27], or arranging for the CAS to provide hints or plans towards proofs of its results [32] or to trust the implementation of CAS while the theorem prover does the book keeping of checking interface de nitions and so on [4] The resulting increase in precision, for example requiring side conditions ....

Harrison, J., and Th ery, L. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In Joyce and Seger [31], pp. 174{ 185.

....developed in several theorem provers as part of a development of theories of real analysis, for example AUTOMATH [dB80] Mizar [Try78] HOL light [Har98] and PVS [Dut96] However such a development does not generally enable us to calculate anything but the simplest integrals. Harrison and Th ery [HT94] experimented with combining such a development with the use of a computer algebra system as an oracle to compute inde nite integrals which were then veri ed by the prover. However this only helps if the computer algebra system gets the integral right 3 The VSDITLU We describe the principle of ....

.... has attracted much research interest recently, in the form of verifying computer algebra algorithms in theorem provers [Th e98] adding inference mechanisms to CAS [CZ94] using proof planning techniques to aid in organising calculations [KKS96] or arranging for provers to make calls to CAS [HT94,HC94], either as oracles for results that are then veri ed or as trusted components for routine manipulations. We have argued elsewhere [Mar98] that while such endeavours are valuable as contributions to theorem proving research the resulting systems are not necessarily widely used by mathematicians as ....

J. Harrison and L. Thery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In Joyce and Seger [JS94], pages 174-185.

....a minor technical problem, but will remain unsolved for the foreseeable future since the complexity (not only the code complexity, but also the mathematical complexity) does not permit a verification of the program itself with currently available program verification methods. In other approaches [HT93] explicit proofs are taken more seriously and computer algebra systems work as an oracle only, receiving a result, whose correctness can then be checked deductively. While this certainly solves the correctness problem, this approach has only a limited coverage, since even checking the correctness ....

J. Harrison and L. Th'ery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In C.-J. H. Seger J. J. Joyce, editor, Higher Order Logic Theorem Proving and its Applications (HUG `93), LNCS 780, pages 174--184. Springer, 1993.

....is often graduate level work . Notice too that theorem provers prove theorems rather than make conjectures or do calculations, and a proof of a simple arithmetical statement, even if performed entirely automatically by means of a suitable tactic, may still be lengthy. Harrison and Th ery [24] experimented with combining such development with the use of Maple as an oracle to compute, for example, factorisations of polynomials or indefinite integrals which could then by verified by the prover. However the success of theorem proving in applications like the verification of hardware has ....

....loss in efficiency if standard algorithms such as factorisation are implemented from first principles. Various hybrid systems have been proposed, where theorem provers call computer algebra systems [26] either using the CAS as an oracle whose results are then then proved in the theorem prover[24], or arranging for the CAS to provide hints or plans towards proofs of its results [30] or to trust the implementation of CAS while the theorem prover does the book keeping of checking interface definitions and so on [3] The resulting increase in precision, for example requiring side conditions ....

Harrison, J., and Th' ery, L. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In Joyce and Seger [29], pp. 174-- 185.

.... Driven by the complexity of real world reasoning problems and practical considerations in designing and interacting with the system, we have seen a rapid move towards integrative frameworks combining various external reasoners [Den93, HKK 94, Dah97] and computation systems [HT93b, BHC95, HT93a, KKS98] Ideally, the reasoning modules in the Omega mega system interact with each other to complete open subgoals during the development of a proof. This can be initiated and supervised on line by the user. This can be also guided by the Omega mega system itself, for instance during proof ....

J. Harrison and L. Th'ery. Extending the HOL Theorem Prover with a Computer Algebra System to Reason About the Reals. In C.-J. H. Seger J. J. Joyce, editor, Higher Order Logic Theorem Proving and its Applications (HUG `93), Volume780 of LNCS, pages 174--184. Springer Verlag, Berlin, 1993.

....the greatest ability to capture mathematical reasoning and respect its spirit. However, they lack some of the qualities of their concurrents. A very promising area of reasearch that has just been scratched is the design of a computer system with reasoning, algorithmic and calculating abilities ([22, 31]) One could imagine a system combining the existing families of systems: proof assistants, theorem provers, computer algebra systems and systems for equational reasoning and rewriting. Ideally, the user of such a system would be free to choose at every moment which component of the system should ....

J.Harrison and L.Thery. Extending the HOL theorem prover with a computer algebra system to reason about the reals, in proceedings of the HOL'93 workshop, 1993.

....have been developed in several theorem provers as part of a development of theories of real analysis, for example AUTOMATH [5] Mizar [37] HOL light [21] and PVS [15] However such a development does not generally enable us to calculate anything but the simplest integrals. Harrison and Thery [24] experimented with combining such a development with the use of a computer algebra system as an oracle to compute indefinite integrals which were then verified by the prover. However this only helps if the computer algebra system gets the integral right 3 The VSDITLU We describe the principle of ....

.... theorem proving has attracted much research interest recently, in the form of verifying computer algebra algorithms in theorem provers [36] adding inference mechanisms to CAS [9] using proof planning techniques to aid in organising calculations [27] or arranging for provers to make calls to CAS [24, 22], either as oracles for results that are then verified or as trusted components for routine manipulations. We have argued elsewhere [32] that while such endeavours are valuable as contributions to theorem proving research the resulting systems are not necessarily widely used by mathematicians as ....

J. Harrison and L. Th'ery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In Joyce and Seger [26], pages 174--185.

....within HOL is extremely difficult, so one way of doing integration is to use a symbolic mathematics package to generate the answer. We do not trust this answer, however, because the result has not been proven within HOL. However, we can use differentiation within HOL to check the answer (Harrison Thery, 1993). We use these ideas in two ways. First, we use a simple auxiliary theorem prover about integers so as to incorporate domain knowledge. Secondly, our tool occasionally uses a heuristic to guess an answer, which is then automatically checked to see whether it is correct. 1.3 Outline Our goal is ....

Harrison, J., & Thery, L. 1993 (Aug.). Extending the HOL Theorem Prover with a Computer Algebra System to Reason about the Reals. In: Proceedings of the HOL User's Group Workshop.

....that can be formally proved by the system combinations is much greater than that provable by MRS alone. However, CAS are very complex programs and therefore only trustworthy to a limited extent, so that the correctness of proofs in such a hybrid system can be questioned. The second category [HT93] is more conscious about the role of proofs, and only uses the CAS as an oracle, receiving a result, whose correctness can then be checked deductively. While this certainly solves the correctness problem, this approach only has a limited coverage, since even checking the correctness of a ....

J. Harrison and L. Th'ery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In C.-J. H. Seger J. J. Joyce, editor, Higher Order Logic Theorem Proving and its Applications (HUG `93), pages 174--184, 1993. Springer-Verlag, LNCS 780.

.... non standard forms of rewriting (here we consider conditional rewriting) At a practical level, we provide the first combination of a symbolic computation system with a theorem prover based on intensional type theory; indeed previous combinations were concerned with extensional type theories ([14, 16]) generic theorem provers ( 3] or systems which are not based on type theory ( 8] The approach followed in this paper was first suggested in [5] and later developed in [7] However, the work of [7] was only of theoretical interest (limited to standard rewriting and without implementation of ....

J. Harrison and L. Thery. Extending the HOL theorem prover with a computer algebra system to reason about the reals. In J.J. Joyce and C-J.H. Seger, editors, Proceedings of HOL'93, volume 780 of Lecture Notes in Computer Science, pages 174--184. Springer-Verlag, 1993.

....choosing one algorithm over another. We will not attempt to compile such an ambitious list, but we will note some good examples. In particular, these include a large number of algorithms commonly associatedwith computer algebra systems. This has been exploited, for example, by Harrison and Thery [16] who physically link the HOL prover and the Maple computer algebra system, performing search and checking in completely different systems. Processes admitting a (relatively) easy checking process include: ffl Factorizing polynomials (or numbers) ffl Finding GCDs of polynomials (or numbers) ffl ....

John Harrison and Laurent Thery, `Extending the HOL theorem prover with a computer algebra system to reason about the reals', In Joyce and Seger [17], pp. 174--184.

....solution of equations. These are difficult, computationally intensive or require sophisticated heuristics. Nevertheless, once a putative answer is found, it is a relatively straightforward matter to check its correctness all that is required for a formal proof. This idea has been explored in [HT93] using a link between HOL and the Maple computer algebra system. More general than using external oracles to produce checkable answers is the idea of delegating all the proof finding to an external proof planner . This idea has been explored by [BvHHS91] 12.3.3 Other optimizations There are a ....

John Harrison and Laurent Th'ery. Extending the hol theorem prover with a computer algebra system to reason about the reals. In Joyce and Seger [JS93b], pages 174--184.

No context found.

J. Harrison, L. Th'ery. Extending HOL Theorem Prover with a Computer Algebra System to Reason About the Reals. In Proceedings of Higher Order Logic Theorem Proving and its Applications. Editors: J.J. Joyce, C. Seger Lecture Notes in Computer Science 780, 1994.

No context found.

J. Harrison and L. Th'ery, Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC