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46
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 ..."
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Cited by 194 (23 self)
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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.
On the Decomposition of the States of SCM
, 1993
"... This article continues the development of the basic terminology ..."
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Cited by 52 (1 self)
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This article continues the development of the basic terminology
Bounding boxes for compact sets inE 2
 Journal of Formalized Mathematics
, 1997
"... Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous ..."
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Cited by 39 (2 self)
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Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous real map from X attains minimum. • Every continuous real map from X attains maximum. Finally, for a compact set in E 2 we define its bounding rectangle and introduce a collection of notions associated with the box.
Constant Assignment Macro Instructions of SCM_FSA. Part II
"... this paper. In this paper m denotes a natural number. We now state two propositions: (1) For every finite sequence p of elements of the instructions of SCMFSA holds dom Load(p) = finsloc(m) : m ! len pg: ..."
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Cited by 15 (5 self)
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this paper. In this paper m denotes a natural number. We now state two propositions: (1) For every finite sequence p of elements of the instructions of SCMFSA holds dom Load(p) = finsloc(m) : m ! len pg:
Minimal Signature for Partial Algebra
 Journal of Formalized Mathematics
, 1995
"... this paper. ..."
The SCMPDS computer and the basic semantics of its instructions
 Journal of Formalized Mathematics
, 1999
"... Summary. The article defines the SCMPDS computer and its instructions. The SCMPDS computer consists of such instructions as conventional arithmetic, “goto”, “return ” and “save instructioncounter ” (“saveIC ” for short). The address used in the “goto ” instruction is an offset value rather than a ..."
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Cited by 12 (10 self)
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Summary. The article defines the SCMPDS computer and its instructions. The SCMPDS computer consists of such instructions as conventional arithmetic, “goto”, “return ” and “save instructioncounter ” (“saveIC ” for short). The address used in the “goto ” instruction is an offset value rather than a pointer in the standard sense. Thus, we don’t define halting instruction directly but define it by “goto 0 ” instruction. The “saveIC ” and “return ” equal almost call and return statements in the usual high programming language. Theoretically, the SCMPDS computer can implement all algorithms described by the usual high programming language including recursive routine. In addition, we describe the execution semantics and halting properties of each instruction.
More on products of many sorted algebras
 Journal of Formalized Mathematics
, 1996
"... Summary. This article is continuation of an article defining products of many sorted algebras [12]. Some properties of notions such as commute, Frege, Args() are shown in this article. Notions of constant of operations in many sorted algebras and projection of products of family of many sorted algeb ..."
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Cited by 11 (0 self)
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Summary. This article is continuation of an article defining products of many sorted algebras [12]. Some properties of notions such as commute, Frege, Args() are shown in this article. Notions of constant of operations in many sorted algebras and projection of products of family of many sorted algebras are defined. There is also introduced the notion of class of family of many sorted algebras. The main theorem states that product of family of many sorted algebras and product of class of family of many sorted algebras are isomorphic.
Conditional Branch Macro Instructions of SCM_FSA. Part II
"... this paper. The following propositions are true: (1) For every state s of SCMFSA holds ..."
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Cited by 11 (4 self)
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this paper. The following propositions are true: (1) For every state s of SCMFSA holds
The construction and computation of conditional statements for SCMPDS
 Journal of Formalized Mathematics
, 1999
"... Summary. We construct conditional statements like the usual high level program language by program blocks of SCMPDS. Roughly speaking, the article justifies such a fact that when the condition of a conditional statement is true (false), and the true (false) branch is shiftable, parahalting and does ..."
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Cited by 9 (5 self)
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Summary. We construct conditional statements like the usual high level program language by program blocks of SCMPDS. Roughly speaking, the article justifies such a fact that when the condition of a conditional statement is true (false), and the true (false) branch is shiftable, parahalting and does not contain any halting instruction, and the false branch is shiftable, then it is halting and its computation result equals that of the true (false) branch. The parahalting means some program halts for all states, this is strong condition. For this reason, we introduce the notions of ”is closed on ” and ”is halting on”. The predicate ”A is closed on B ” denotes program A is closed on state B, and ”A is halting on B ” denotes program A is halting on state B. We obtain a similar theorem to the above fact by replacing parahalting by ”is closed on ” and ”is halting on”.