Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals.

Abstract. Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom’s Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples. 1 Overview and Related Work Decision methods for nonlinear real arithmetic are essential to the formal verification of cyber-physical systems and formalized mathematics. Classically, thes

A combination of geometry theorem proving and nonstandard analysis with application to Newton's Pricipia

SIGLEAvailable from British Library Document Supply Centre-DSC:8723.247(469) / BLDSC - British Library Document Supply CentreGBUnited Kingdo

A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia

SIGLEAvailable from British Library Document Supply Centre-DSC:8723.247(442) / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Formalizing Integration Theory, with an Application to Probabilistic Algorithms

Inter alia, Lebesgue-style integration plays a major role in advanced probability. We formalize a significant part of its theory in Higher Order Logic using the generic interactive theorem prover Isabelle/Isar. This involves concepts of elementary measure theory, real-valued random variables as Borelmeasurable functions, and a stepwise inductive definition of the integral itself. Building on previous work about formal verification of probabilistic algorithms, we exhibit an example application in this domain; another primitive for randomized functional programming is developed to this end. All proofs are carried out in human readable style using the Isar language

Formal Proof of a Wave Equation Resolution Scheme: the Method Error

Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization of the simplest one and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical pen-and-paper proofs. To our knowledge, this is the first time such kind of mathematical proof is machine-checked.Comment: This paper has been withdrawn by the authors. Please refere to arXiv:1005.082