Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
Similarity of Formulae
-
Agata Darmochwal
-
Warsaw University
-
Andrzej Trybulec
-
Warsaw University, Bialystok
Summary.
-
The main objective of the paper is to define the concept of the similarity
of formulas. We mean by similar formulas the two formulas that differs only
in the names of bound variables. Some authors (compare [14])
call such
formulas {\em congruent}. We use the word {\em similar} following
[12], [11], [13].
The concept is unjustfully
neglected in many logical handbooks. It is intuitively quite clear,
however the exact
definition is not simple. As far as we know, only W.A.~Pogorzelski and
T.~Prucnal [13]
define it in the precise way.
We follow basically the Pogorzelski's definition (compare [12]).
We define renumeration of
bound variables and we say that two formulas are similar if after renumeration
are equal. Therefore we need a rule of chosing bound variables independent of
the original choice. Quite obvious solution is to use
consecutively variables $x_{k+1},x_{k+2},\dots$, where $k$ is the maximal
index
of free variable occurring in the formula. Therefore after the renumeration
we get the new formula in which different quantifiers bind different
variables.
It is the reason that the result of renumeration applied to
a formula $\varphi$ we call {\em $\varphi$ with variables separated}.
The terminology and notation used in this paper have been
introduced in the following articles
[19]
[6]
[24]
[22]
[17]
[20]
[25]
[26]
[4]
[5]
[16]
[3]
[10]
[23]
[21]
[2]
[18]
[15]
[1]
[9]
[7]
[8]
Contents (PDF format)
Bibliography
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Functions and their basic properties.
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Functions from a set to a set.
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A classical first order language.
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Czeslaw Bylinski and Grzegorz Bancerek.
Variables in formulae of the first order language.
Journal of Formalized Mathematics,
1, 1989.
- [10]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
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- [11]
Wlodzimierz Lesisz and Witold A. Pogorzelski.
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- [12]
Witold A. Pogorzelski.
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- [15]
Piotr Rudnicki and Andrzej Trybulec.
A first order language.
Journal of Formalized Mathematics,
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- [16]
Andrzej Trybulec.
Binary operations applied to functions.
Journal of Formalized Mathematics,
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- [17]
Andrzej Trybulec.
Domains and their Cartesian products.
Journal of Formalized Mathematics,
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Andrzej Trybulec.
Semilattice operations on finite subsets.
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Andrzej Trybulec.
Function domains and Fr\aenkel operator.
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Andrzej Trybulec.
Subsets of real numbers.
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Addenda, 2003.
- [23]
Andrzej Trybulec and Agata Darmochwal.
Boolean domains.
Journal of Formalized Mathematics,
1, 1989.
- [24]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [25]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [26]
Edmund Woronowicz.
Relations defined on sets.
Journal of Formalized Mathematics,
1, 1989.
Received November 22, 1991
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