Results 1  10
of
680
The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
Abstract

Cited by 722 (70 self)
 Add to MetaCart
(Show Context)
Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on [9] and [10].
Finite Sequences and Tuples of Elements of a Nonempty Sets
, 1990
"... this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D ..."
Abstract

Cited by 330 (7 self)
 Add to MetaCart
(Show Context)
this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D
Basis of Real Linear Space
, 1990
"... this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G ..."
Abstract

Cited by 287 (21 self)
 Add to MetaCart
(Show Context)
this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G
Pigeon hole principle
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the notion of a predicate that states that a function is onetoone at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given ..."
Abstract

Cited by 270 (13 self)
 Add to MetaCart
(Show Context)
Summary. We introduce the notion of a predicate that states that a function is onetoone at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given element of the range of the finite sequence, the substring preceding (and succeeding) the first occurrence of given element of the range. At the end of the article we prove the pigeon hole principle.
The Modification of a Function by a Function and the Iteration of the Composition of a Function
, 1990
"... ..."
Topological Properties of Subsets in Real Numbers
 JOURNAL OF FORMALIZED MATHEMATICS
, 2002
"... ..."
The sum and product of finite sequences of real numbers
 Journal of Formalized Mathematics
, 1990
"... ..."
(Show Context)
Monotone Real Sequences. Subsequences
"... this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasin ..."
Abstract

Cited by 97 (9 self)
 Add to MetaCart
(Show Context)
this paper. We follow the rules: n, m, k are natural numbers, r is a real number, and s 1 , s 2 , s 3 are sequences of real numbers. Let s 1 be a partial function from N to R. We say that s 1 is increasing if and only if: (Def. 1) For every n holds s 1 (n) ! s 1 (n + 1): We say that s 1 is decreasing if and only if: (Def. 2) For every n holds s 1 (n + 1) ! s 1 (n): We say that s 1 is nondecreasing if and only if: (Def. 3) For every n holds s 1 (n) s 1 (n + 1): We say that s 1 is nonincreasing if and only if: (Def. 4) For every n holds s 1 (n + 1) s 1 (n): Let f be a function. We say that f is constant if and only if: (Def. 5) For all sets n 1 , n 2 such that n 1 2 dom f and n 2 2 domf holds f(n 1 ) = f(n 2 ): Let us consider s 1 . Let us observe that s 1 is constant if and only if: (Def. 6) There exists r such that for every n holds s 1 (n) = r:
Combining of Circuits
, 2002
"... this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S ..."
Abstract

Cited by 95 (25 self)
 Add to MetaCart
this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S
An Overview of the MIZAR Project
 UNIVERSITY OF TECHNOLOGY, BASTAD
, 1992
"... The Mizar project is a longterm effort aimed at developing software to support a working mathematician in preparing papers. A. Trybulec, the leader of the project, has designed a language for writing formal mathematics. The logical structure of the language is based on a natural deduction system ..."
Abstract

Cited by 94 (1 self)
 Add to MetaCart
(Show Context)
The Mizar project is a longterm effort aimed at developing software to support a working mathematician in preparing papers. A. Trybulec, the leader of the project, has designed a language for writing formal mathematics. The logical structure of the language is based on a natural deduction system developed by Ja'skowski. The texts written in the language are called Mizar articles and are organized into a data base. The TarskiGrothendieck set theory forms the basis of doing mathematics in Mizar. The implemented processor of the language checks the articles for logical consistency and correctness of references to other articles.