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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.
Baire’s Category Theorem and Some Spaces Generated from Real Normed Space 1
"... Summary. As application of complete metric space, we proved a Baire’s category theorem. Then we defined some spaces generated from real normed space and discussed each of them. In the second section, we showed the equivalence of convergence and the continuity of a function. In other sections, we sho ..."
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Cited by 2 (2 self)
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Summary. As application of complete metric space, we proved a Baire’s category theorem. Then we defined some spaces generated from real normed space and discussed each of them. In the second section, we showed the equivalence of convergence and the continuity of a function. In other sections, we showed some topological properties of two spaces, which are topological space and linear topological space generated from real normed space.
The Product Space of Real Normed Spaces and its Properties
"... Summary. In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces. ..."
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Cited by 1 (1 self)
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Summary. In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces.
Partial Differentiation on Normed Linear Spaces R n
"... Summary. In this article, we define the partial differentiation of functions of real variable and prove the linearity of this operator [18]. ..."
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Summary. In this article, we define the partial differentiation of functions of real variable and prove the linearity of this operator [18].