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tps: A theorem proving system for classical type theory
 Journal of Automated Reasoning
, 1996
"... This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a comb ..."
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Cited by 71 (6 self)
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This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems which TPS can prove completely automatically are given to illustrate certain aspects of TPS’s behavior and problems of theorem proving in higherorder logic. 7
Automatic Concept Formation in Pure Mathematics
"... The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best ..."
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Cited by 38 (28 self)
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The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best first search. This approachhasledHRtothediscoveryofinterestingnewmathematics and enables it to build theories from just the axioms of finite algebras.
Solving Open Questions and Other Challenge Problems Using Proof Sketches
, 2001
"... . In this article, we describe a set of procedures and strategies for searching for proofs in logical systems based on the inference rule condensed detachment. The procedures and strategies rely on the derivation of proof sketchessequences of formulas that are used as hints to guide the search for ..."
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Cited by 29 (14 self)
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. In this article, we describe a set of procedures and strategies for searching for proofs in logical systems based on the inference rule condensed detachment. The procedures and strategies rely on the derivation of proof sketchessequences of formulas that are used as hints to guide the search for sound proofs. In the simplest case, a proof sketch consists of a subproofkey lemmas to prove, for exampleand the proof is completed by lling in the missing steps. In the more general case, a proof sketch consists of a sequence of formulas sucient to nd a proof, but it may include formulas that are not provable in the current theory. We nd that even in this more general case, proof sketches can provide valuable guidance in nding sound proofs. Proof sketches have been used successfully for numerous problems coming from a variety of problem areas. We have, for example, used proof sketches to nd several new twoaxiom systems for Boolean algebra using the Sheer stroke. Keywords: proof sk...
TPS: A TheoremProving System for Classical Type Theory
, 1996
"... . This is description of TPS, a theoremproving system for classical type theory (Church's typed #calculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a comb ..."
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Cited by 16 (0 self)
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. This is description of TPS, a theoremproving system for classical type theory (Church's typed #calculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems that TPS can prove completely automatically are given to illustrate certain aspects of TPS's behavior and problems of theorem proving in higherorder logic. AMS Subject Classification: 0304, 68T15, 03B35, 03B15, 03B10. Key words: higherorder logic, type theory, mating, connection, expansion proof, natural deduction. 1. Introduction TPS is a theoremproving system for classical type theory ## (Church's typed #calculus [20]) which has been under development at Carnegie Mellon University for a number years. This paper gives a general...
The growth of mathematical knowledge: an open world view
 The growth of mathematical knowledge, Kluwer, Dordrecht 2000
"... mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but ..."
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Cited by 5 (5 self)
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mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past ” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself ” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks
The Creation and Use of a Knowledge Base of Mathematical Theorems and Definitions
, 1995
"... IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or ..."
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Cited by 2 (2 self)
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IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or rewrite rules. Because there is more easilyaccessible information in a sequent than there is in the formula it represents, a simple algorithm can be used to search the knowledge base for the most useful theorem or definition to be used in the theoremproving process. This paper describes how the sequents in the knowledge base are formed from theorems stated by the user and how the knowledge base is used in the theoremproving process. An example of a theorem proved and the English proof output are also given. 1 Introduction The motivating goal behind this work is to develop a theoremproving system which will be useful to both an expert and a nonexpert in the attempt to prove theorems in ...
A Framework for Using Knowledge in Tableau Proofs
 Proc. International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, PontMousson
, 1997
"... . The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau cal ..."
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Cited by 2 (1 self)
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. The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau calculus. A natural, wellmotivated and simple restriction (implemented in IPR) to the method provides a powerful framework for the automation of the selection of theorems from a knowledge base for use in theorem proving. The restrictions are related to semantic resolution restrictions and the setofsupport restriction in resolution, and to hypertableaux and the weak connection condition in tableaux. We also present additional tableau rules used by the IPR prover for handling some equality which is not complete but is sufficient for handling the problems in its intended domain of problem solving. 1 Introduction The rules presented in this paper allow an automatic theorem proving pro...
Discoveries and Experiments in the Automation of Mathematical Reasoning
, 2002
"... vii List of Figures xii Chapter 1. ..."
Premise Selection in the Naproche System
"... Abstract. Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. ..."
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Abstract. Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. In this paper, we describe an axiom selection method for series of related problems that is based on logical and textual proximity and tries to mimic a human way of understanding mathematical texts. We present first results that indicate that this approach is indeed useful. Key words: formal mathematics, automated theorem proving, axiom selection 1