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

Perfect Numbers in ACL2

A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 < 1 + 2 + 3 + 4 + 6 is not perfect. An ACL2 theory of perfect numbers is developed and used to prove, in ACL2(r), this bit of mathematical folklore: Even if there are infinitely many perfect numbers the series of the reciprocals of all perfect numbers converges.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

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

Some congruence properties of three well-known sequences: Two notes

AbstractThe three sequences mentioned in the title are Ramanujan's τ-function, the coefficients cn of Klein, Fricke, and Shimura, and the sequence an of Apèry numbers. In the first note, it is shown that cn ≡ τ(n)(mod 11). In the second note it is shown that for a prime p, ap+1 ≡ 25 + 60p(mod p2)

Vivez sans temps morts, jouissez sans entraves: Language and Identity in the May \u2768 Student-Worker Action Committees

This paper attempts to recast and distance the interpretation of May \u2768 away from popular historiographical trends in which the movement is treated as a generational revolt or a cultural revolution that allowed France to develop into a modern, liberal society. The character of the relationship between militant students and workers remains the most complicated and understudied element of the uprising. Thus, the actions taken by student and worker militants at the occupied Sorbonne constitute the most radical element of the movement, and represented the only cases in which the radical language produced by activist students achieved form, namely in the organization of effective student-worker cooperation in the university and factory occupations in Paris

Jim Renacci

This is an advertisement for the re-election of Jim Renacci to the U.S. House of Representatives