超演绎原理

Paul Brauner, C. Houtmann, C. Kirchner
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引用次数: 48

摘要

在谓词逻辑中,一个定理P在一个理论Th中成立的证明通常是用自然演绎或序贯演算的方式进行的,使用理论中包含的所有信息以统一的方式进行。十年前引入的演绎模允许我们利用理论Th的计算部分来进行演绎的真正计算模。本文以相继演算为中心,提出并研究了对偶概念,并运用对偶理论系统、正确、完整地用新的演绎法规则丰富了演绎法体系。我们称这种新的扣除制度为“超级扣除”。在经典序演学的基础上,介绍了一种证明项语言和切消过程。在适当而自然的假设下证明了强归一化,从而保证了嵌入理论与演绎系统的一致性。在这种新体系中得到的证明更接近于人类的直觉和实践。因此,我们概述了如何使用超演绎和演绎模来建立新的可扩展证明助手的形式基础,如lemuridae,我们的超演绎模的原型实现。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Principles of Superdeduction
In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focusing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system "superdeduction ". We introduce a proof-term language and a cut-elimination procedure both based on Christian Urban's work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently sketch how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants like lemuridae, our prototypal implementation of superdeduction modulo.
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