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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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Implementing an Efficient Theorem Prover
, 2003
"... Since the invention of resolution and paramodulation calculi, theoretical research in the area of resolutionbased theorem proving has achieved a remarkable progress in constructing inference systems based on various refinements of these calculi. The developed theory has a significant but not yet fu ..."
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Since the invention of resolution and paramodulation calculi, theoretical research in the area of resolutionbased theorem proving has achieved a remarkable progress in constructing inference systems based on various refinements of these calculi. The developed theory has a significant but not yet fully realised potential for such applications as formal hardware and software development, computer algebra, and assisting human mathematicians. One of the main obstacles to creating a practically valuable technology on the base of the theory is the lack of knowledge of efficient implementation techniques. Attempts of implementing efficient theorem provers constitute the main source of such knowledge. In this thesis we give an account of such an attempt. We anatomise the design and implementation of our resolution and superpositionbased theorem prover Vampire whose performance has led it to first places in international competitions during the last four years. A high level description of the overall design of the system and the underlying principles is followed by a detailed discussion of a number of key implementation techniques, devised in the course of our study. These include (i) Limited Resource Strategy, a novel adaptive approach to organising saturation process in the presence of a time limit
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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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.
The Difficulties of Definite Integration
"... Indefinite integration is the inverse operation to differentiation, and, before we can understand what we mean by indefinite integration, we need to understand what we mean by differentiation. 1.1 What is differentiation? 1. An analytic operation: f ′ (x0) = limx→x0 f(x)−f(x0) x−x0 2. An algebraic ..."
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Indefinite integration is the inverse operation to differentiation, and, before we can understand what we mean by indefinite integration, we need to understand what we mean by differentiation. 1.1 What is differentiation? 1. An analytic operation: f ′ (x0) = limx→x0 f(x)−f(x0) x−x0 2. An algebraic operation, satisfying (a + b) ′ = a ′ + b ′ , (ab) ′ = a ′ b + b ′ a, x ′ = 1. We note the different ways in which the fact that we mean “differentiation with respect to x ” is expressed in the two formulations. 1.1.1 Two interpretations of atan Analytically, atan(x) = y: y = tan(x) and −π/2 < y < π/2. Algebraically, atan(f) ′ ′ f = 1+f 2. Therefore only defined “up to a constant”. Analytically, c is a constant iff c(x1) = c(x2)∀x1, x2, and, in this view, the Heaviside function is not a constant. Algebraically, c is a constant iff c ′ = 0, and, in this view, the Heaviside function is a constant. ∗ The author is grateful to many people for their discussions of definite integration, notably Andrews Adams, Jacques Carrette, Tony Hearn and Daniel Lichtblau. 1 1.1.2 Comparing the two approaches • The success of computer algebra is that one can model the first by the second. • The problem of computer algebra is that the model, particularly when it comes to inverse differentiation and the handling of “up to a constant”, is not perfect. The first deals with functions (say R ↦ → R), the second with algebraic expressions.
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 to at ..."
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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.
Digital Look Up Tables and Real Number Theorem Proving
, 2001
"... We consider the utility of digital look up tables, as adjuncts/helpers to computer algebra systems. The requirements for dealing with logical side conditions raised by such tables are considered and proposals for using theorem proving technology as black box aids are considered. In addition, the ..."
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We consider the utility of digital look up tables, as adjuncts/helpers to computer algebra systems. The requirements for dealing with logical side conditions raised by such tables are considered and proposals for using theorem proving technology as black box aids are considered. In addition, the use of real number theorem proving libraries to support validation of table entries is also presented.
Computer Algebra and Automated Reasoning
"... eorem of calculus: Given a function f and a function F such that: F 0 = f; and given two limits a; b such that [a; b] Dom(f) and f is continuous on [a; b] then b R a f(x)dx = F (b) F (a) While there are tricks to avoid problems with discontinuities computer algebra systems in general igno ..."
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eorem of calculus: Given a function f and a function F such that: F 0 = f; and given two limits a; b such that [a; b] Dom(f) and f is continuous on [a; b] then b R a f(x)dx = F (b) F (a) While there are tricks to avoid problems with discontinuities computer algebra systems in general ignore many discontinuities and deal poorly with most others. It is usually left to the user (sometimes without informing them of this fact) that they must manually check the side conditions on using the fundamental theorem of calculus. In particular, computer algebra systems deal very poorly with calcuations involving parameters. In cases where they will correctly identify (and sometimes correctly work a