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Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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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 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
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 45 (17 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
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 17 (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...
HigherOrder Quantification and Proof Search
 In Proceedings of the AMAST confrerence, LNCS
, 2002
"... Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because firstorder logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these a ..."
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Cited by 7 (4 self)
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Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because firstorder logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program.
Solving for Set Variables in HigherOrder Theorem Proving
 Proceedings of the 18th International Conference on Automated Deduction
, 2002
"... In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. U ..."
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Cited by 6 (1 self)
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In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. Using the KnasterTarski Fixed Point Theorem [ 15 ] , constraints whose solutions require recursive de nitions can be solved as xed points of monotone set functions. In this paper, we consider an approach to higherorder theorem proving which intertwines conventional theorem proving in the form of mating search with generating and solving set constraints.
Encoding the Calculus of Constructions in a HigherOrder Logic
 IN EIGHTH ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1993
"... We present an encoding of the calculus of constructions (CC) in a higherorder intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, we demonstrate a direct correspondence between proofs in these t ..."
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Cited by 4 (2 self)
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We present an encoding of the calculus of constructions (CC) in a higherorder intuitionistic logic (I) in a direct way, so that correct typing in CC corresponds to intuitionistic provability in a sequent calculus for I. In addition, we demonstrate a direct correspondence between proofs in these two systems. The logic I is an extension of hereditary Harrop formulas (hh) which serve as the logical foundation of the logic programming language Prolog. Like hh, I has the uniform proof property, which allows a complete nondeterministic search procedure to be described in a straightforward manner. Via the encoding, this search procedure provides a goal directed description of proof checking and proof search in CC.
The Calculus of Constructions as a Framework for Proof Search with Set Variable Instantiation
, 2000
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Proof Search with Set Variable Instantiation in the Calculus of Constructions
 Automated Deduction: CADE13, volume 1104 of Lecture Notes in Arti Intelligence
, 1996
"... . We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higherorder logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of firstorder lo ..."
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Cited by 2 (1 self)
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. We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higherorder logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of firstorder logic that allows existential quantification over set variables. The method finds maximal solutions for this special class of higherorder variables. This class of variables can also be identified in CC. The existence of a correspondence between higherorder logic and higherorder type theories such as CC is wellknown. CC can be viewed as an extension of higherorder logic where the basic terms of the language, the simplytyped terms, are replaced with terms containing dependent types. We adapt Bledsoe's procedure to the corresponding class of variables in CC and extend it to handle terms with dependent types. 1 Introduction Both higherorder logic and higherorder type theories serve as th...
SET VARIABLES
"... ABSTRACT: A procedure is described that gives values to set variables in automatic theorem proving. The result is that a theorem is thereby reduced to first order logic, which is often much easier to prove. This procedure handles a part of higher order logic, a small but important part. It is not as ..."
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Cited by 1 (0 self)
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ABSTRACT: A procedure is described that gives values to set variables in automatic theorem proving. The result is that a theorem is thereby reduced to first order logic, which is often much easier to prove. This procedure handles a part of higher order logic, a small but important part. It is not as general as the methods of Huet, Andrews, Pietrzykowski, and Haynes and Henschen, but it seems to be much faster when it applies. It is more in the spirit of J.L. Darlington's FMatching. This procedure is not domain specific: results have been obtained In intermediate analysis (the intermediate value theorem), topology, logic, and program verification (finding internal assertions). This method is a "maximal method" in that a largest (or maximal) set is usually produced if there is one. A preliminary version has been programmed for the computer and run to prove several theorems.