Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999
Association of Mizar Users
Defining by Structural Induction in the Positive Propositional Language
-
Andrzej Trybulec
-
University of Bialystok
Summary.
-
The main goal of the paper consists in proving schemes for defining
by structural induction in the language defined by Adam Grabowski
[11]. The article consists of four parts. Besides the
preliminaries where we prove some simple facts still missing in the library,
they are:
\item{-} ``About the language'' in which the consequences of the fact that the algebra
of formulae is free are formulated,
\item{-} ``Defining by structural induction'' in which two schemes are proved,
\item{-} ``The tree of the subformulae'' in which a scheme proved in the previous
section is used to define the tree of subformulae; also some simple
facts about the tree are proved.
The terminology and notation used in this paper have been
introduced in the following articles
[14]
[10]
[17]
[16]
[1]
[12]
[18]
[3]
[9]
[13]
[8]
[4]
[15]
[2]
[5]
[6]
[7]
[11]
-
Preliminaries
-
About the Language
-
Defining by Structural Induction
-
The Tree of the Subformulae
Bibliography
- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
Journal of Formalized Mathematics,
1, 1989.
- [2]
Grzegorz Bancerek.
Introduction to trees.
Journal of Formalized Mathematics,
1, 1989.
- [3]
Grzegorz Bancerek.
The ordinal numbers.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Grzegorz Bancerek.
Cartesian product of functions.
Journal of Formalized Mathematics,
3, 1991.
- [5]
Grzegorz Bancerek.
K\"onig's Lemma.
Journal of Formalized Mathematics,
3, 1991.
- [6]
Grzegorz Bancerek.
Sets and functions of trees and joining operations of trees.
Journal of Formalized Mathematics,
4, 1992.
- [7]
Grzegorz Bancerek.
Joining of decorated trees.
Journal of Formalized Mathematics,
5, 1993.
- [8]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
Journal of Formalized Mathematics,
1, 1989.
- [9]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [10]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
- [11]
Adam Grabowski.
Hilbert positive propositional calculus.
Journal of Formalized Mathematics,
11, 1999.
- [12]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
Journal of Formalized Mathematics,
1, 1989.
- [13]
Beata Padlewska.
Families of sets.
Journal of Formalized Mathematics,
1, 1989.
- [14]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [15]
Andrzej Trybulec.
Many-sorted sets.
Journal of Formalized Mathematics,
5, 1993.
- [16]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
- [17]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [18]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received April 23, 1999
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