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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

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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
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
Structuring of ComputerGenerated Proofs by Cut Introduction
"... . As modern Automated Deduction systems rely heavily on the use of a machineoriented representation of a given problem, together with sophisticated redundancyavoiding techniques, a major task in convincing human users of the correctness of automatically generated proofs is the intelligible rep ..."
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Cited by 1 (0 self)
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. As modern Automated Deduction systems rely heavily on the use of a machineoriented representation of a given problem, together with sophisticated redundancyavoiding techniques, a major task in convincing human users of the correctness of automatically generated proofs is the intelligible representation of these proofs. In this paper, we propose the use of the cutrule in the humanoriented presentation of computergenerated proofs. The intelligent application of cuts enables the integration of essential lemmata and therefore shortens and structures proof presentation. We show that many translation techniques in Automated Deduction, such as antiprenexing and some forms of normal form translations, can be described as cuts and are indeed part of the deductive solution of a problem. Furthermore, we demonstrate the connection between symmetric simplification, quantorial extension principles and the application of the cutrule. 1 Introduction Most of today's Automated Ded...