Cauchy Mean Theorem

• Autorzy: Adam Grabowski
• Języki publikacji: EN

Abstrakty

EN

The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Słowa kluczowe

EN

geometric mean; arithmetic mean; AM-GM inequality; Cauchy mean theorem

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 26A06: One-variable calculus
• 26A12: Rate of growth of functions, orders of infinity, slowly varying functions

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2014
• Tom: 22
• Numer: 2
• Strony: 157-166

Daty

otrzymano

2014-06-13

online

2015-02-05

Twórcy

autor

Adam Grabowski

• Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok Poland

Bibliografia

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Identyfikatory

DOI

10.2478/forma-2014-0016