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PROOFS IN HIGHERORDER LOGIC
, 1983
"... Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either ..."
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Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ETproof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higherorder logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cutelimination theorem for higherorder logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what
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 70 (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 61 (11 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.
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 ..."
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Cited by 21 (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...
Expansion Tree Proofs and Their Conversion to Natural Deduction Proofs
 in 7th International Conference on Automated Deduction, edited by
"... Abstract: We present a new form of Herbrand's theorem which is centered around structures called expansion trees. Such trees contains substitution formulas and selected (critical) variables at various nonterminal nodes. These trees encode a shallow formula and a deep formula the latter contai ..."
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Abstract: We present a new form of Herbrand's theorem which is centered around structures called expansion trees. Such trees contains substitution formulas and selected (critical) variables at various nonterminal nodes. These trees encode a shallow formula and a deep formula the latter containing the formulas which label the terminal nodes of the expansion tree. If a certain relation among the selected variables of an expansion tree is acyclic and if the deep formula of the tree is tautologous, then we say that the expansion tree is a special kind of proof, called an ETproof, of its shallow formula. Because ETproofs are sufficiently simple and general (expansion trees are, in a sense, generalized formulas), they can be used in the context of not only firstorder logic but also a version of higherorder logic which properly contains firstorder logic. Since the computational logic literature has seldomly dealt with the nature of proofs in higherorder logic, our investigation of ETproofs will be done entirely in this setting. It can be shown that a formula has an ETproof if and only if that formula is a theorem of higherorder logic. Expansion trees have several pleasing practical and theoretical properties. To demonstrate this fact, we use ETproofs to extend and complete Andrews ' procedure [41 for automatically constructing natural deductions proofs. We shall also show how to use a mating for an ETproof's tautologous, deep formula to provide this procedure with the "look ahead " needed to determine if certain lines are unnecessary to prove other lines and when and how backchaining can be done. The resulting natural deduction proofs are generally much shorter and more readable than proofs build without using this mating information. This conversion process works without needing any search. Details omitted in this paper can be found in the author's dissertation [161.
Automation of HigherOrder Logic
 THE HANDBOOK OF THE HISTORY OF LOGIC, EDS. D. GABBAY & J. WOODS; VOLUME 9: LOGIC AND COMPUTATION, EDITOR JÖRG SIEKMANN
, 2014
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TPS User’s Manual
, 2011
"... conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National ..."
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conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National
TPS: An Interactive and Automatic Tool for Proving Theorems of Type Theory
 Higher Order Logic Theorem Proving and Its Applications: 6th International Workshop, HUG '93, volume 780 of Lecture Notes in Computer Science
, 1994
"... This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed lcalculus). TPS can be used 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 deduc ..."
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This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed lcalculus). TPS can be used 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. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 lcalculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are humanreadable formal proofs. An example of such a proof which was produced aut...