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Induction and decision procedures.

Deepak Kapur, Jürgen Giesl, Mahadevan Subramaniam (2004)

RACSAM

Mechanization of inductive reasoning is an exciting research area in artificial intelligence and automated reasoning with many challenges. An overview of our work on mechanizing inductive reasoning based on the cover set method for generating induction schemes from terminating recursive function definitions and using decision procedures is presented. This paper particularly focuses on the recent work on integrating induction into decision procedures without compromising their automation.

Integral of Complex-Valued Measurable Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Introduction to Diophantine Approximation

Yasushige Watase (2015)

Formalized Mathematics

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

Introduction to Formal Preference Spaces

Eliza Niewiadomska, Adam Grabowski (2013)

Formalized Mathematics

In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative...

Introduction to Liouville Numbers

Adam Grabowski, Artur Korniłowicz (2017)

Formalized Mathematics

The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and [...] It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is...

Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Hiroshi Yamazaki, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2013)

Formalized Mathematics

In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

Isomorphisms of Direct Products of Finite Commutative Groups

Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama (2013)

Formalized Mathematics

We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups

Isomorphisms of Direct Products of Finite Cyclic Groups

Kenichi Arai, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

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