Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

**Andrzej Trybulec**- University of Bialystok

- 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.

- Preliminaries
- About the Language
- Defining by Structural Induction
- The Tree of the Subformulae

- [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.

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