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63
Function domains and Frænkel operator
 Journal of Formalized Mathematics
, 1990
"... Summary. We deal with a non–empty set of functions and a non–empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B A is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some th ..."
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Cited by 147 (18 self)
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Summary. We deal with a non–empty set of functions and a non–empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B A is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some theorems concerning these concepts are proved, as well as a number of schemes dealing with infinity and the Axiom of Choice. The article contains a number of schemes allowing for simple logical transformations related to terms constructed with the Frænkel Operator.
The sum and product of finite sequences of real numbers.
 Formalized Mathematics,
, 1990
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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 ..."
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Cited by 93 (25 self)
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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
Binary operations applied to finite sequences.
 Formalized Mathematics,
, 1990
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The product and the determinant of matrices with entries in a field
, 2003
"... Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field. ..."
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Cited by 27 (0 self)
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Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field.
Semigroup operations on finite subsets
 Journal of Formalized Mathematics
, 1990
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Function Domains and Fraenkel Operator
, 1990
"... this paper. In this paper A, B are non empty sets and X is a set. In this article we present several logical schemes. The scheme Fraenkel5' concerns a non empty set A; a unary functor F yielding a set, and two unary predicates P ; Q; and states that: ..."
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Cited by 22 (6 self)
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this paper. In this paper A, B are non empty sets and X is a set. In this article we present several logical schemes. The scheme Fraenkel5' concerns a non empty set A; a unary functor F yielding a set, and two unary predicates P ; Q; and states that:
The formalization of simple graphs
 Journal of Formalized Mathematics
, 1994
"... Summary. A graph is simple when • it is nondirected, • there is at most one edge between two vertices, • there is no loop of length one. A formalization of simple graphs is given from scratch. There is already an article [10], dealing with the similar subject. It is not used as a startingpoint, be ..."
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Cited by 21 (0 self)
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Summary. A graph is simple when • it is nondirected, • there is at most one edge between two vertices, • there is no loop of length one. A formalization of simple graphs is given from scratch. There is already an article [10], dealing with the similar subject. It is not used as a startingpoint, because [10] formalizes directed nonempty graphs. Given a set of vertices, edge is defined as an (unordered) pair of different two vertices and graph as a pair of a set of vertices and a set of edges. The following concepts are introduced: • simple graph structure, • the set of all simple graphs, • equality relation on graphs. • the notion of degrees of vertices; the number of edges connected to, or the number of adjacent vertices, • the notion of subgraphs, • path, cycle, • complete and bipartite complete graphs, Theorems proved in this articles include: • the set of simple graphs satisfies a certain minimality condition, • equivalence between two notions of degrees.