Results 1  10
of
294
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
The Modification of a Function by a Function and the Iteration of the Composition of a Function
, 1990
"... ..."
Manysorted sets
 Journal of Formalized Mathematics
, 1993
"... Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is def ..."
Abstract

Cited by 198 (23 self)
 Add to MetaCart
(Show Context)
Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is defined as ∀i∈Ixi ∈ Xi. I was prompted by a remark in a paper by Tarlecki and Wirsing: “Throughout the paper we deal with manysorted sets, functions, relations etc.... We feel free to use any standard settheoretic notation without explicit use of indices ” [6, p. 97]. The aim of this work was to check the feasibility of such approach in Mizar. It works. Let us observe some peculiarities: empty set (i.e. the many sorted set with empty set of indices) belongs to itself (theorem 133), we get two different inclusions X ⊆ Y iff ∀i∈IXi ⊆ Yi and X ⊑ Y iff ∀xx ∈ X ⇒ x ∈ Y equivalent only for sets that yield non empty values. Therefore the care is advised.
A classical first order language
 Journal of Formalized Mathematics
, 1990
"... this paper. In this paper i, j, k are natural numbers. Let x, y, a, b be sets. The functor (x = y a,b) yields a set and is defined as follows: (Def. 1) (x = y a,b) = a, if x = y, b, otherwise. Let D be a non empty set, let x, y be sets, and let a, b be elements of D. Then (x = y a,b) is an e ..."
Abstract

Cited by 154 (0 self)
 Add to MetaCart
(Show Context)
this paper. In this paper i, j, k are natural numbers. Let x, y, a, b be sets. The functor (x = y a,b) yields a set and is defined as follows: (Def. 1) (x = y a,b) = a, if x = y, b, otherwise. Let D be a non empty set, let x, y be sets, and let a, b be elements of D. Then (x = y a,b) is an element of D
The sum and product of finite sequences of real numbers
 Journal of Formalized Mathematics
, 1990
"... ..."
(Show Context)
Many sorted algebras
 Journal of Formalized Mathematics
, 1994
"... Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between ..."
Abstract

Cited by 130 (14 self)
 Add to MetaCart
(Show Context)
Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between (1 sorted) universal algebras [8] and many sorted algebras with one sort only is described by introducing two functors mapping one into the other. The construction is done this way that the composition of both functors is the identity on universal algebras.
Directed sets, nets, ideals, filters, and maps
 Journal of Formalized Mathematics
, 1996
"... Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8 ..."
Abstract

Cited by 117 (31 self)
 Add to MetaCart
(Show Context)
Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8, p. 2]), the definition of ideals and filters and the definition of maps preserving arbitrary and directed sups and arbitrary and filtered infs (1.9 also in [8, p. 4]). The concepts of semilattices, supsemiletices and poset lattices (1.8 in [8, p. 4]) are also introduced. A number of facts concerning the above notion and including remarks 1.4, 1.5, and 1.10 from [8, pp. 3–5] is presented.
A Borsuk theorem on homotopy types
 Journal of Formalized Mathematics
, 1991
"... Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retrac ..."
Abstract

Cited by 108 (7 self)
 Add to MetaCart
(Show Context)
Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.
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