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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.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs ..."
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Cited by 5 (1 self)
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational t ..."
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Cited by 5 (0 self)
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
Trees and EhrenfeuchtFraïssé games
, 1997
"... We study trees T of height at most omega1 with no uncountable branches, and their applications in the study of pairs (A,B) of nonisomorphic structures over a fixed vocabulary. There is a natural quasiordering of such trees in terms of the existence of a strictly increasing mapping from one tree t ..."
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Cited by 4 (3 self)
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We study trees T of height at most omega1 with no uncountable branches, and their applications in the study of pairs (A,B) of nonisomorphic structures over a fixed vocabulary. There is a natural quasiordering of such trees in terms of the existence of a strictly increasing mapping from one tree to another. We investigate in depth the structure of this quasiordering and relate its properties to properties of pairs (A,B) of structures. Many new constructions of pairs of highly equivalent nonisomorphic structures are given.
Towards Formal Support for Generic Programming
, 2003
"... der EberhardKarlsUniversität Tübingen für das Fach Informatik ..."
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der EberhardKarlsUniversität Tübingen für das Fach Informatik
Tarski Grothendieck Set Theory
 Journal of Formalized Mathematics, Axiomatics
, 1989
"... this paper x, y, z, u, N , M , X, Y , Z denote sets. The following proposition is true (2) 1 If for every x holds x # X i# x # Y, then X = Y. Let us consider y. The functor {y} is defined by: (Def. 1) x # {y} i# x = y. Let us consider z. The functor {y, z} is defined as follows: (Def. ..."
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this paper x, y, z, u, N , M , X, Y , Z denote sets. The following proposition is true (2) 1 If for every x holds x # X i# x # Y, then X = Y. Let us consider y. The functor {y} is defined by: (Def. 1) x # {y} i# x = y. Let us consider z. The functor {y, z} is defined as follows: (Def. 2) x # {y, z} i# x = y or x = z. Let us observe that the functor {y, z} is commutative. Let us consider X, Y . The predic
Noname manuscript No. (will be inserted by the editor) Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
"... the date of receipt and acceptance should be inserted later Abstract Smart premise selection is essential when using automated reasoning as a tool for largetheory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning t ..."
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the date of receipt and acceptance should be inserted later Abstract Smart premise selection is essential when using automated reasoning as a tool for largetheory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learningbased premise selection in two ways. First, a newly available minimal dependency analysis of existing highlevel formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATPbased reverification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 largetheory mathematical problems is constructed, extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE stateoftheart system for automated reasoning in large theories. 1