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Nine Chapters of Analytic Number Theory in Isabelle/HOL

Manuel Eberl, Michael Wagner
2019 International Conference on Interactive Theorem Proving  
Prime Number Theorem The asymptotics of arithmetical functions such as the prime ω function, the divisor count σ0(n), and Euler's totient function ϕ(n) 2012 ACM Subject Classification Mathematics of computing  ...  In this paper, I present a formalisation of a large portion of Apostol's Introduction to Analytic Number Theory in Isabelle/HOL.  ...  Introduction The formalisation of Apostol's book in Isabelle/HOL started from the simple desire to have more properties about Euler's ϕ function available in the system.  ... 
doi:10.4230/lipics.itp.2019.16 dblp:conf/itp/Eberl19 fatcat:362y54hm75cmjjnykukb3u4a7i

Mathematical Method and Proof

Jeremy Avigad
2006 Synthese  
On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem.  ...  I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.  ...  Proofs of Fermat's theorem, Lagrange's theorem, and Wilson's theorem can be found in almost any elementary textbook on number theory; see for example [31, 42] . 11 Euler's proof assumed a slightly weaker  ... 
doi:10.1007/s11229-005-4064-5 fatcat:4ienrswtvzbetaw74oxyu7omzi

A Formal Proof of the Minor-Exclusion Property for Treewidth-Two Graphs [chapter]

Christian Doczkal, Guillaume Combette, Damien Pous
2018 Lecture Notes in Computer Science  
Euler's theorem was formalized in Mizar [15] . To the best of our knowledge, the theory of minors and tree decompositions was never formalized.  ...  There are more formalizations in other interactive theorem provers. For instance, planar graphs were formalized in Isabelle/HOL for the Flyspeck project [16] .  ...  Cite this paper as: Doczkal C., Combette G., Pous D.: A Formal Proof of the Minor-Exclusion Property for Treewidth-Two Graphs. In: Avigad J., Mahboubi A. (eds) Interactive Theorem Proving. ITP 2018.  ... 
doi:10.1007/978-3-319-94821-8_11 fatcat:ckzgr3fcarehzl4fufiftjbs4q

Mizar in a Nutshell

Adam Grabowski
unpublished
ACKNOWLEDGMENT The authors wish to express their sincere appreciation to the outstanding work of Andrzej Trybulec, the inventor of Mizar and the leader of the Mizar project.  ...  As of now, also 53 facts from the Wiedijk's "Top 100 Mathematical Theorems" are formalized 1 , such as Fundamental Theorem of Algebra, Euler's Polyhedron Formula, Ramsey's Theorem, Sylow's theorems, and  ...  However, to state that a certain family of subsets is in fact a partition one needs to make a reference to the corresponding definitional theorem.  ... 
fatcat:htmhjetpubgwzbspup5a46tflm

Foundations of scientific research (Foundations of Research Activities) [article]

N. M. Glazunov
2012 arXiv   pre-print
During years 2008 to 2011 author gives several courses on Foundations of Scientific Research at Computer Science Faculty of the National Aviation University in Kiev.  ...  Some sections of the text contain enough material to lectures, but in some cases these are sketchs without references to Foundations of Research Activities.  ...  Some fundamental notions of the theory of Petri nets can be described in Mizar formalism. The Mizar Language is a formal language derived from the mathematical vernacular.  ... 
arXiv:1212.1651v1 fatcat:e3oe5neztff5vjlj4trchbi3qa

Primality Testing is Polynomial-time: A Mechanised Verification of the AKS Algorithm [article]

Hing Lun Joseph Chan, University, The Australian National
2019
The entire work is carried out in the HOL4 theorem prover. The correctness of the AKS algorithm relies on a main theorem developed by the AKS team, based on the theory of finite fields.  ...  For the formal proof, we need to dive into several advanced topics in finite field, in particular the existence and uniqueness of finite fields, and properties of cyclotomic polynomials.  ...  (Euler's Totient Theorem).  ... 
doi:10.25911/5f58b06ca124e fatcat:objty3blpbbyha3evgeduua64a