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14
A proofproducing decision procedure for real arithmetic
 Automated deduction – CADE20. 20th international conference on automated deduction
, 2005
"... Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate conv ..."
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Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious lowlevel proofs. We can identify several very common limitations of such procedures: Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination. Often they are not complete even for the supported class of formulas; in particular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y) They seldom handle nontrivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods. Many of the procedures are standalone decision algorithms that produce no certificate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems arising in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning over
A HOL theory of Euclidean space
 In Hurd and Melham [7
, 2005
"... Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in ..."
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Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in a simple and useful way, even though HOL does not permit dependent types. In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, e.g. checking compatibility in matrix multiplication. Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary Ndimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions. 1 1 The problem with R N
Formalizing an analytic proof of the prime number theorem
 Journal of Automated Reasoning
"... describe the computer formalization of a complexanalytic proof of the Prime Number Theorem (PNT), a classic result from number theory characterizing the asymptotic density of the primes. The formalization, conducted using the HOL Light theorem prover, proceeds from the most basic axioms for mathema ..."
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describe the computer formalization of a complexanalytic proof of the Prime Number Theorem (PNT), a classic result from number theory characterizing the asymptotic density of the primes. The formalization, conducted using the HOL Light theorem prover, proceeds from the most basic axioms for mathematics yet builds from that foundation to develop the necessary analytic machinery including Cauchy’s integral formula, so that we are able to formalize a direct, modern and elegant proof instead of the more involved ‘elementary ’ ErdösSelberg argument. As well as setting the work in context and describing the highlights of the formalization, we analyze the relationship between the formal proof and its informal counterpart and so attempt to derive some general lessons about the formalization of mathematics.
Linear quantifier elimination
 In Automated reasoning (IJCAR), volume 5195 of LNCS
, 2008
"... Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formula ..."
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Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection). 1
Primality Proving with Elliptic Curves
"... de recherche ISSN 02496399 ISRN INRIA/RR6155FR+ENG ..."
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de recherche ISSN 02496399 ISRN INRIA/RR6155FR+ENG
A formalized proof of Dirichlet’s theorem on primes in arithmetic progression
"... We describe the formalization using the HOL Light theorem prover of Dirichlet’s theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a startingpoint. The goal of this paper is ..."
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We describe the formalization using the HOL Light theorem prover of Dirichlet’s theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a startingpoint. The goal of this paper is twofold. First we describe a simple and efficient proof of the theorem informally, which is otherwise difficult to find in one selfcontained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days. 1.
A Methodology for the Formal Verification of FFT Algorithms in HOL
 In Formal Methods in ComputerAided Design, LNCS 3312
, 2004
"... Abstract. This paper addresses the formal specification and verification of fast Fourier transform (FFT) algorithms at different abstraction levels based on the HOL theorem prover. We make use of existing theories in HOL on real and complex numbers, IEEE standard floatingpoint, and fixedpoint ar ..."
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Abstract. This paper addresses the formal specification and verification of fast Fourier transform (FFT) algorithms at different abstraction levels based on the HOL theorem prover. We make use of existing theories in HOL on real and complex numbers, IEEE standard floatingpoint, and fixedpoint arithmetics to model the FFT algorithms. Then, we derive, by proving theorems in HOL, expressions for the accumulation of roundoff error in floating and fixedpoint FFT designs with respect to the corresponding ideal real and complex numbers specification. The HOL formalization and proofs are found to be in good agreement with the theoretical paperandpencil counterparts. Finally, we use a classical hierarchical proof approach in HOL to prove that the FFT implementations at the register transfer level (RTL) implies the corresponding high level fixedpoint algorithmic specification. 1
Parametric linear arithmetic over ordered fields in Isabelle/HOL
"... We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the non ..."
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We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the nonparametric case. The formalization is based on axiomatic type classes and automatically carries over to e.g. the rational, real and nonstandard real numbers. It is executable, can be applied to HOL formulae by reflection and performs well on practical examples.
Error Analysis and Verification of an IEEE 802.11 OFDM Modem using Theorem Proving 1
"... IEEE 802.11 is a widely used technology which powers many of the digital wireless communication revolutions currently taking place. It uses OFDM (Orthogonal Frequency Division Multiplexing) in its physical layer which is an efficient way to deal with multipath, good for relatively slow timevarying ..."
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IEEE 802.11 is a widely used technology which powers many of the digital wireless communication revolutions currently taking place. It uses OFDM (Orthogonal Frequency Division Multiplexing) in its physical layer which is an efficient way to deal with multipath, good for relatively slow timevarying channels, and robust against narrowband interference. In this paper, we formally specify and verify an implementation of the IEEE 802.11 standard physical layer based OFDM modem using the HOL (Higher Order Logic) theorem prover. The versatile expressive power of HOL helped us model the original design at all abstraction levels starting from a floatingpoint model to the fixedpoint design and then synthesized and implemented in FPGA technology. We have been able to find a bug in one of the blocks of the design that is responsible for modulation which implementation diverts from the constellation provided in the IEEE standard specification. The paper also derives new expressions for the rounding error accumulated during ideal real to floatingpoint and fixedpoint transitions at the algorithmic level and performs a formal error analysis for