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On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
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Cited by 13 (3 self)
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Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
Bounded Domains and Unbounded Domains
, 2003
"... this paper. 1. DEFINITIONS OF BOUNDED DOMAIN AND UNBOUNDED DOMAIN We follow the rules: m, n are natural numbers, r, s are real numbers, and x, y are sets. We now state several propositions: (1) If r 0, then r =r ..."
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this paper. 1. DEFINITIONS OF BOUNDED DOMAIN AND UNBOUNDED DOMAIN We follow the rules: m, n are natural numbers, r, s are real numbers, and x, y are sets. We now state several propositions: (1) If r 0, then r =r
A Proof of the Jordan Curve Theorem via the Brouwer Fixed Point Theorem
"... Abstract – The aim of the paper is to report on MIZAR codification of the Jordan curve theorem, a theorem chosen as a challenge to be completely verified using formal methods at the time when they started being commonly used. Formalization was done based on proofs taken from the literature, where th ..."
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Abstract – The aim of the paper is to report on MIZAR codification of the Jordan curve theorem, a theorem chosen as a challenge to be completely verified using formal methods at the time when they started being commonly used. Formalization was done based on proofs taken from the literature, where theorems mentioned in the title of the paper from ”Brouwer’s Fixed Point Theorem and the