Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989
Association of Mizar Users
Introduction to Trees
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Grzegorz Bancerek
-
Warsaw University, Bialystok
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Partially supported by Le Hodey Foundation.
The part of this work had been done on Mizar Workshop '89 (Fourdrain, France)
in Summer '89.
Summary.
-
The article consists of two parts: the first one deals with the concept of
the prefixes of a finite sequence, the second one introduces and deals with
the concept of tree. Besides some auxiliary propositions concerning
finite sequences are presented. The trees are introduced as non-empty sets
of finite sequences of natural numbers which are closed on prefixes and
on sequences of less numbers (i.e. if $\langle n_1$, $n_2$, $\dots$,
$n_k\rangle$ is a vertex (element) of a tree and $m_i \leq n_i$
for $i = 1$, $2$, $\dots$, $k$, then $\langle m_1$, $m_2$, $\dots$,
$m_k\rangle$ also is). Finite trees, elementary trees with $n$ leaves,
the leaves and the subtrees of a tree, the inserting of a tree into another
tree, with a node used for determining the place of insertion,
antichains of prefixes, and height and width of finite trees
are introduced.
MML Identifier:
TREES_1
The terminology and notation used in this paper have been
introduced in the following articles
[6]
[8]
[2]
[7]
[9]
[4]
[3]
[5]
[1]
Contents (PDF format)
Bibliography
- [1]
Grzegorz Bancerek.
Cardinal numbers.
Journal of Formalized Mathematics,
1, 1989.
- [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
Journal of Formalized Mathematics,
1, 1989.
- [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [5]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
1, 1989.
- [6]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [7]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
- [8]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [9]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received October 25, 1989
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