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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 69 (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
Natural Deduction as HigherOrder Resolution
 Journal of Logic Programming
, 1986
"... An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. ..."
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Cited by 54 (8 self)
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An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.
The TPS theorem proving system
 9th International Conference on Automated Deduction, Argonne, Illinois
, 1988
"... TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be ..."
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Cited by 27 (5 self)
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TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be used to search for an expansion proof [10, 11] of a theorem, which represents in a nonredtmdant way the basic combinatorial information required to construct a proof of
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 ..."
Abstract

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...
Presenting intuitive deductions via symmetric simplification
 In CADE10: Proceedings of the tenth international conference on Automated deduction
, 1990
"... In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented pro ..."
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Cited by 15 (4 self)
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In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented proofs. Previously known translation procedures, though complete, tend to produce unintuitive deductions. One of the major flaws in such procedures is that too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof. We present an algorithm, symmetric simplification, for discovering useful lemmas in deductions of theorems in first and higherorder logic. This algorithm, which has been implemented in the TPS system, has the feature that resulting deductions may no longer have the weak subformula property. It is currently limited, however, in that it only generates lemmas of the form C ∨ ¬C ′ , where C and C ′ have the same negation normal form. 1
Integrating TPS and ΩMEGA
 JOURNAL OF UNIVERSAL COMPUTER SCIENCE
, 1999
"... This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control ov ..."
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Cited by 7 (4 self)
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This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in Tps; in interactive mode, all features of the Tpssystem are available to the user. If the subproblem which is passed to Tps contains concepts defined in Ωmega's database of mathematical theories, these definitions are not instantiated but are also passed to Tps. Using a special theory which contains proof tactics that model the NDcalculus variant of Tps within mega, any complete or partial proof generated in Tps can be translated one to one into an mega proof plan. Proof transformation is realised by proof plan expansion in Ωmega's 3dimensional proof data structure, and remains transparent to the user.
Integrating TPS with ΩMEGA
, 1998
"... We report on the integration of Tps as an external reasoning component into the mathematical assistant system Ωmega. Thereby Tps can be used both as an automatic theorem prover for higher order logic as well as interactively employed from within the Ωmega environment. Tps proofs can be directly inc ..."
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Cited by 2 (1 self)
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We report on the integration of Tps as an external reasoning component into the mathematical assistant system Ωmega. Thereby Tps can be used both as an automatic theorem prover for higher order logic as well as interactively employed from within the Ωmega environment. Tps proofs can be directly incorporated into Ωmega on a tactic level enabling their visualization and verbalization. Using an example we show how Tps proofs can be inserted into Ωmega's knowledge base by expanding them to calculus level using both Ωmega's tactic mechanism and the first order theorem prover Otter. Furthermore we demonstrate how the facts from Ωmega's knowledge base can be used to build a Tps library.
Proof transformations from searchoriented into interactionoriented tableau calculi
 Journal of Universal Computer Science
, 1999
"... Abstract: Logic calculi, and Gentzentype calculi in particular, can be categorised into two types: searchoriented and interactionoriented calculi. Both these types have certain inherentcharacteristics stemming from the purpose for which they are designed. In this paper, we give a general characte ..."
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Cited by 1 (0 self)
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Abstract: Logic calculi, and Gentzentype calculi in particular, can be categorised into two types: searchoriented and interactionoriented calculi. Both these types have certain inherentcharacteristics stemming from the purpose for which they are designed. In this paper, we give a general characterisation of the two types and present two calculi that are typical representatives of their respective class. We introduce a method for transforming proofs in the searchoriented calculus into proofs in the interactionoriented calculus, and we demonstrate that the di culties arising with devising such a transformation do not pertain to the speci c calculi we have chosen as examples but are general. We also give examples for the application of our transformation procedure.
Automated theorem proving: mapping logic into AI
 Proceedings of the International Symposium on Methodologies for Intelligent Systems
, 1986
"... ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "formal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not alway ..."
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Cited by 1 (0 self)
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ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "formal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not always the methodologies, ATP has been considered to be within the domain of AI. We explore the methodologies of ATP, including the logics that underlie the theorem provers, and discuss some of the mechanisms that utilize these logics. These include term rewriting systems, mathematical induction, inductionless induction and even mixed integer programming. The ATP field, via resolution, has even provided the foundation for an exciting AI and database programming language, PROLOG. We conclude with a new method for extending the PROLOG system to work with nonHorn clause sets within a positive logic format, particularly simple for "slightly nonHorn " clause sets.