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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
Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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Cited by 21 (5 self)
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
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...
Nivelle. Automated proof construction in type theory using resolution
 Special Issue Mechanizing and Automating Mathematics: In honour of N.G. de Bruijn
, 2002
"... Abstract. We provide techniques to integrate resolution logic with equality in type theory. The results may be rendered as follows. − A clausification procedure in type theory, equipped with a correctness proof, all encoded using higherorder primitive recursion. − A novel representation of clauses ..."
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Abstract. We provide techniques to integrate resolution logic with equality in type theory. The results may be rendered as follows. − A clausification procedure in type theory, equipped with a correctness proof, all encoded using higherorder primitive recursion. − A novel representation of clauses in minimal logic such that the λrepresentation of resolution steps is linear in the size of the premisses. − A translation of resolution proofs into lambda terms, yielding a verification procedure for those proofs. − The power of resolution theorem provers becomes available in interactive proof construction systems based on type theory. 1.
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
Uniform Variable Splitting
 In Contributions to the Doctoral Programme of the Second International Joint Conference on Automated Reasoning (IJCAR 2004
, 2007
"... ost with b. The two occurrences are variable independent. In a calculus with universal variables this is easily recognized, but there are cases where a variable is not universal, but still independent of many other occurrences of the same variable. Now, let us reverse the order of rule application s ..."
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ost with b. The two occurrences are variable independent. In a calculus with universal variables this is easily recognized, but there are cases where a variable is not universal, but still independent of many other occurrences of the same variable. Now, let us reverse the order of rule application such that the inference is below the inference. If the calculus introduces a new free variable with every inference, then the derivations are variablepure [13]. This is exemplified in (b), where the above variable independence is revealed due to the di#erence in inference order. In order to have goaldirected search and keep a tight relation to matrix systems [15, 13, 7], it is desirable to have invariance under order of rule application, which is not a property enjoyed by variablepure calculi. (The leaf sequents in (a) di#er from those in (b).) To obtain this invariance, one can employ a way of reusing free variables. If the di#erent occurrences of the same formula introduce the sa
Redirecting proofs by contradiction
"... This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algori ..."
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This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algorithm is implemented in Isabelle’s Sledgehammer, where it enhances the legibility of machinegenerated proofs. 1
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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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.