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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, ..."
Abstract

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
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...