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.

MML Identifier: HILBERT2

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]

Contents (PDF format)

  1. Preliminaries
  2. About the Language
  3. Defining by Structural Induction
  4. 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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