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24
MBase: Representing Knowledge and Context for the Integration of Mathematical Software Systems
, 2000
"... In this article we describe the data model of the MBase system, a webbased, ..."
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Cited by 41 (11 self)
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In this article we describe the data model of the MBase system, a webbased,
Proving Equalities in a Commutative Ring Done Right in Coq
 Theorem Proving in Higher Order Logics (TPHOLs 2005), LNCS 3603
, 2005
"... We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The e#ciency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement e#cient algorithms while kee ..."
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Cited by 25 (0 self)
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We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The e#ciency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement e#cient algorithms while keeping the complexity of the correctness proofs low.
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Cited by 19 (2 self)
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
Controlling Control Systems: An Application of Evolving Retrenchment
"... We review retrenchment as a liberalisation of refinement, for the description of applications too rich (e.g. using continuous and infinite types) for refinement. A specialisation of the notion, evolving retrenchment is introduced, motivated by the need for an approximate, evolving notion of simu ..."
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Cited by 17 (12 self)
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We review retrenchment as a liberalisation of refinement, for the description of applications too rich (e.g. using continuous and infinite types) for refinement. A specialisation of the notion, evolving retrenchment is introduced, motivated by the need for an approximate, evolving notion of simulation. The focus of the paper is the case study, a substantial secondorder linear control system. The design step from continuous to zeroorder hold discrete system is expressible as an evolving retrenchment. Thus we demonstrate that the retrenchment approach can formalise the development of useful applications, which are outside the scope of refinement. The work is presented in a data typeenriched language containing the B language of J.R. Abrial. 1
Fast Reflexive Arithmetic Tactics the linear case and beyond
 in "Types for Proofs and Programs (TYPES’06)", Lecture Notes in Computer Science
, 2006
"... Abstract. When goals fall in decidable logic fragments, users of proofassistants expect automation. However, despite the availability of decision procedures, automation does not come for free. The reason is that decision procedures do not generate proof terms. In this paper, we show how to design ef ..."
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Cited by 15 (6 self)
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Abstract. When goals fall in decidable logic fragments, users of proofassistants expect automation. However, despite the availability of decision procedures, automation does not come for free. The reason is that decision procedures do not generate proof terms. In this paper, we show how to design efficient and lightweight reflexive tactics for a hierarchy of quantifierfree fragments of integer arithmetics. The tactics can cope with a wide class of linear and nonlinear goals. For each logic fragment, offtheshelf algorithms generate certificates of infeasibility that are then validated by straightforward reflexive checkers proved correct inside the proofassistant. This approach has been prototyped using the Coq proofassistant. Preliminary experiments are promising as the tactics run fast and produce small proof terms. 1
Certifying solutions to permutation group problems
 In F. Baader, ed, CADE19, LNAI 2741
, 2003
"... Abstract. We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to pr ..."
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Cited by 13 (0 self)
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Abstract. We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to provide a full proof of correctness of the answer. To guarantee correctness we use proof planning techniques, which construct proofs in a humanoriented reasoning style. This gives the human mathematician the necessary insight into the computed solution, as well as making it feasible to check the solution for relatively large groups. 1
An Overview of A Formal Framework For Managing Mathematics
 Annals of Mathematics and Artificial Intelligence
, 2003
"... Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform t ..."
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Cited by 12 (6 self)
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Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system.
Nontrivial Symbolic Computations in Proof Planning
 In Proc. of FroCoS 2000, LNCS 1794
, 2000
"... We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using ..."
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Cited by 6 (3 self)
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We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using a powerful computer algebra system during the planning process to do nontrivial symbolic computations. Results of these computations are checked during the refinement of a proof plan to a calculus level proof using a small, selfimplemented, system that gives us protocol information on its calculation. This protocol can be easily expanded into a checkable lowlevel calculus proof ensuring the correctness of the computation. We demonstrate our approach with the concrete implementation in the Omega system.
Context aware calculation and deduction  Ring equalities via Gröbner Bases in Isabelle
 TOWARDS MECHANIZED MATHEMATICAL ASSISTANTS (CALCULEMUS AND MKM 2007), LNAI
, 2007
"... We address some aspects of a proposed system architecture for mathematical assistants, integrating calculations and deductions by common infrastructure within the Isabelle theorem proving environment. Here calculations may refer to arbitrary extralogical mechanisms, operating on the syntactic struc ..."
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Cited by 6 (5 self)
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We address some aspects of a proposed system architecture for mathematical assistants, integrating calculations and deductions by common infrastructure within the Isabelle theorem proving environment. Here calculations may refer to arbitrary extralogical mechanisms, operating on the syntactic structure of logical statements. Deductions are devoid of any computational content, but driven by procedures external to the logic, following to the traditional “LCF system approach”. The latter is extended towards explicit dependency on abstract theory contexts, with separate mechanisms to interpret both logical and extralogical content uniformly. Thus we are able to implement proof methods that operate on abstract theories and a range of particular theory interpretations. Our approach is demonstrated in Isabelle/HOL by a proofprocedure for generic ring equalities via Gröbner Bases.
Trustable Communication Between Mathematics Systems
 IN PROC. OF CALCULEMUS 2003
, 2003
"... This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client inter ..."
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Cited by 4 (3 self)
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This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client interface is linked to a server interface by an asymmetric connection consisting of a pair of translations. Answers to requests are trustable in the sense that they are correct provided a small set of prescribed conditions are satis ed. The framework is robust with respect to interface extension and can process requests for abstract services, where the server interface is not fully specified.