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}.

#### MML Identifier: CQC_SIM1

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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Received November 22, 1991

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