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Computer algebra meets automated theorem proving: Integrating Maple and pvs
 Theorem Proving in Higher Order Logics (TPHOLs 2001), volume 2152 of LNCS
, 2001
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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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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.
Automated Theorem Proving in Support of Computer Algebra: Symbolic Definite Integration as a Case Study
"... We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integrat ..."
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We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integration, and present a verifiable symbolic definite integral table look up: a system which matches a query comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of lemmas about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon Fateman's webbased table by including parametric limits of integration and queries w...
Finding Polynomial Invariants for Imperative Loops in the Theorema System
, 2006
"... Abstract. We present an algorithm for finding valid polynomial relations (i. e. invariants) among program variables for imperative loops. The algorithm is implemented in the verification environment for imperative programs (using Hoare logic) in the frame of the Theorema system (www.theorema.org). W ..."
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Abstract. We present an algorithm for finding valid polynomial relations (i. e. invariants) among program variables for imperative loops. The algorithm is implemented in the verification environment for imperative programs (using Hoare logic) in the frame of the Theorema system (www.theorema.org). We use techniques from (polynomial) algebra and combinatorics, namely Gröbner Bases, variable elimination, algebraic dependencies and symbolic summation (the Gosper algorithm, handling geometric series, Cfinite solving). These methods are demonstrated on several examples which have been treated completely automatically by our implementation.
Equational Prover of Theorema
, 2003
"... The equational prover of the Theorema system is described. It is implemented on Mathematica and is designed for unit equalities in the first order or in the applicative higher order form. A (restricted) usage of sequence variables and Mathematica builtin functions is allowed. ..."
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Cited by 7 (6 self)
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The equational prover of the Theorema system is described. It is implemented on Mathematica and is designed for unit equalities in the first order or in the applicative higher order form. A (restricted) usage of sequence variables and Mathematica builtin functions is allowed.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every e ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
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
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
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Hidden Verification for Computational Mathematics
"... We present hidden verification as a means to make the power of computational logic available to users of computer algebra systems while shielding them fi'om its complexity. We have implemented in PVS a library of facts about elementary and transcendental functions, and automatic procedures ..."
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Cited by 3 (1 self)
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We present hidden verification as a means to make the power of computational logic available to users of computer algebra systems while shielding them fi'om its complexity. We have implemented in PVS a library of facts about elementary and transcendental functions, and automatic procedures to attempt proofs of continuity, convergence and differentiability for functions in this class. These are called directly from Maple by a simple pipelined interface. Hence we are able to support the analysis of differential equations in Maple by direct calls to PVS for: result refinement and verification, discharge of verification conditions, harnesses to ensure more reliable differential equation solvers, and verifiable lookup tables.
System Description: Interface between Theorema And External Automated Deduction Systems
 In Linton and Sebastiani [175
, 2001
"... The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface a ..."
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The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface allows combining external systems with each other as well as with \internal" Theorema provers.