6 research outputs found
Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent
The verification of many algorithms for calculating transcendental functions
is based on polynomial approximations to these functions, often Taylor series
approximations. However, computing and verifying approximations to the
arctangent function are very challenging problems, in large part because the
Taylor series converges very slowly to arctangent-a 57th-degree polynomial is
needed to get three decimal places for arctan(0.95). Medina proposed a series
of polynomials that approximate arctangent with far faster convergence-a
7th-degree polynomial is all that is needed to get three decimal places for
arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness
and convergence rate of this sequence of polynomials. The proof is particularly
beautiful, in that it uses many results from real analysis. Some of these
necessary results were proven in prior work, but some were proven as part of
this effort.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Equivalence of the Traditional and Non-Standard Definitions of Concepts from Real Analysis
ACL2(r) is a variant of ACL2 that supports the irrational real and complex
numbers. Its logical foundation is based on internal set theory (IST), an
axiomatic formalization of non-standard analysis (NSA). Familiar ideas from
analysis, such as continuity, differentiability, and integrability, are defined
quite differently in NSA-some would argue the NSA definitions are more
intuitive. In previous work, we have adopted the NSA definitions in ACL2(r),
and simply taken as granted that these are equivalent to the traditional
analysis notions, e.g., to the familiar epsilon-delta definitions. However, we
argue in this paper that there are circumstances when the more traditional
definitions are advantageous in the setting of ACL2(r), precisely because the
traditional notions are classical, so they are unencumbered by IST limitations
on inference rules such as induction or the use of pseudo-lambda terms in
functional instantiation. To address this concern, we describe a formal proof
in ACL2(r) of the equivalence of the traditional and non-standards definitions
of these notions.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Fourier Series Formalization in ACL2(r)
We formalize some basic properties of Fourier series in the logic of ACL2(r),
which is a variant of ACL2 that supports reasoning about the real and complex
numbers by way of non-standard analysis. More specifically, we extend a
framework for formally evaluating definite integrals of real-valued, continuous
functions using the Second Fundamental Theorem of Calculus. Our extended
framework is also applied to functions containing free arguments. Using this
framework, we are able to prove the orthogonality relationships between
trigonometric functions, which are the essential properties in Fourier series
analysis. The sum rule for definite integrals of indexed sums is also
formalized by applying the extended framework along with the First Fundamental
Theorem of Calculus and the sum rule for differentiation. The Fourier
coefficient formulas of periodic functions are then formalized from the
orthogonality relations and the sum rule for integration. Consequently, the
uniqueness of Fourier sums is a straightforward corollary.
We also present our formalization of the sum rule for definite integrals of
infinite series in ACL2(r). Part of this task is to prove the Dini Uniform
Convergence Theorem and the continuity of a limit function under certain
conditions. A key technique in our proofs of these theorems is to apply the
overspill principle from non-standard analysis.Comment: In Proceedings ACL2 2015, arXiv:1509.0552
Implementing an Automatic Differentiator in ACL2
The foundational theory of differentiation was developed as part of the
original release of ACL2(r). In work reported at the last ACL2 Workshop, we
presented theorems justifying the usual differentiation rules, including the
chain rule and the derivative of inverse functions. However, the process of
applying these theorems to formalize the derivative of a particular function is
completely manual. More recently, we developed a macro and supporting functions
that can automate this process. This macro uses the ACL2 table facility to keep
track of functions and their derivatives, and it also interacts with the macro
that introduces inverse functions in ACL2(r), so that their derivatives can
also be automated. In this paper, we present the implementation of this macro
and related functions.Comment: In Proceedings ACL2 2011, arXiv:1110.447
Foundational Certification of Code Transformations Using Automatic Differentiation
Automatic Differentiation (AD) is concerned with the semantics augmentation of an input program representing a function to form a transformed program that computes the function's derivatives. To ensure the correctness of the AD transformed code, particularly for safety critical applications, we aim at certifying the algebraic manipulations at the heart of the AD process. We have considered a WHILE-language and have shown how such proofs can be constructed by using an appropriate relational Hoare logic.In particular, we have shown how such inference rules can be constructed for both the forward and reverse mode AD by using an abductive logical reasoning
Formalization of Real Analysis: A Survey of Proof Assistants and Libraries
International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis