Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

On Same Equivalents of Well-foundedness

Piotr Rudnicki
University of Alberta, Edmonton
Andrzej Trybulec
Warsaw University, Bialystok

Summary.

Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending $\omega$-chains) have been proved in Mizar and the proofs were mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. This work was inspired by T.~Franzen's paper ~[14]. Franzen's proofs were written by a mathematician having an argument with a computer scientist. We were curious about the effort needed to formalize Franzen's proofs given the state of the Mizar Mathematical Library at that time (July 1996). The formalization went quite smoothly once the mathematics was sorted out.

This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

MML Identifier: WELLFND1

The terminology and notation used in this paper have been introduced in the following articles [19] [12] [23] [21] [2] [24] [9] [16] [25] [11] [10] [18] [3] [5] [4] [13] [1] [22] [20] [8] [17] [6] [15] [7]

Contents (PDF format)

1. Preliminaries
2. Well Founded Relational Structures

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