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Comprehension and Description in Tableaux
, 1997
"... Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure fi ..."
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Various approaches have been invented for enabling an automated theorem proving program to find proofs in set theory. The present approach is completely automatic and quite successful on many problems which are showcased as challenge problems for provers in set theory. In fact, this procedure finds proofs of several of these examples without search. We implement the comprehension schema by means of tableau reduction and expansion rules. We also discuss the implementation of the definite descriptor in tableaux and special rules for handling equality effectively and in a tractable way in set theory. 1 Introduction An inference rule that "builds in" set theory at the inference level is the objective of Research Problem 8. More precisely, just as the employment of paramodulation permits one to avoid using any equality axioms other than reflexivity, the soughtafter inference rule for set theory would permit one to avoid using a number of the axioms in Godel's approach. L...
A Framework for the Creation and Use of a Knowledge Base of Mathematical Theorems and Definitions
, 1995
"... A mathematician knows thousands of theorems and definitions and is able to choose just the needed results and use them at just the right time in the theoremproving process. The problem of codifying some bit of this process is the topic of this paper. IPR is an automatic theoremproving system i ..."
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A mathematician knows thousands of theorems and definitions and is able to choose just the needed results and use them at just the right time in the theoremproving process. The problem of codifying some bit of this process is the topic of this paper. IPR is an automatic theoremproving system intended particularly for use in mathematics. It discovers the proofs of theorems in mathematics by applying known theorems and definitions from a knowledge base. Theorems and definitions are stored in the knowledge base in the form of "sequents" rather than formulas or rewrite rules. The sequentsinto which a theorem is reduced before being put into the knowledge base consistently mirror the ways a human might use the theorem. Because the data are in this form, natural fetching algorithms can be used to search the knowledge base for the most useful theorem to be used in the theoremproving process. 1 Introduction The theory presented here, implemented in IPR, provides a natur...