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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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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
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"... Abstract. This paper is part of an ongoing effort to examine the role of extensionality in higherorder logic and provide tools for analyzing higherorder calculi. In an earlier paper, we have presented eight classes of higher order models with respect to various combinations of Boolean extensionali ..."
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Abstract. This paper is part of an ongoing effort to examine the role of extensionality in higherorder logic and provide tools for analyzing higherorder calculi. In an earlier paper, we have presented eight classes of higher order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we have developed a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of higherorder calculi with respect to these model classes. This framework, employs a strong saturation criterion which prevents analysis of, e.g., the deductive power of machineoriented calculi. In this paper we extend our saturated abstract consistency approach and obtain analogous model existence results without assuming saturation. For this, we replace the saturation
\OmegaAnts A Blackboard Architecture for the Integration of Reasoning Techniques into Proof Planning
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Model Existence for Higher Order Logic
, 2000
"... In this paper we provide a semantical metatheory that will support the development of higherorder calculi for automated theorem proving like the corresponding methodology has in rstorder logic. To reach this goal, we establish classes of models that adequately characterize the existing theorempr ..."
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In this paper we provide a semantical metatheory that will support the development of higherorder calculi for automated theorem proving like the corresponding methodology has in rstorder logic. To reach this goal, we establish classes of models that adequately characterize the existing theoremproving calculi and we present a standard methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of machineoriented calculi with respect to this model classes. 1 Motivation In classical rstorder predicate logic, it is rather simple to assess the deductive power of a calculus: rstorder logic has a wellestablished and intuitive settheoretic semantics, relative to which, completeness can easily be veried using for instance the abstract consistency method (cf. the introductory textbooks [And86, Fit90]). This wellunderstood metatheory has supported the development of calculi adapted to special applications { such a...