Abstract
Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules. When only terms are rewritten, cut admissibility in those systems is equivalent to the confluence of the rewriting system, as shown by Dowek, RTA 2003, LNCS 2706. This is no longer true when considering rewriting rules involving propositions. In this paper, we show that, in the same way that it is possible to recover confluence using Knuth-Bendix completion, one can regain cut admissibility in the general case using standard saturation techniques. This work relies on a view of proposition rewriting rules as oriented clauses, like term rewriting rules can be seen as oriented equations. This also leads us to introduce an extension of deduction modulo with conditional term rewriting rules.
Keywords
- Inference Rule
- Proof System
- Natural Deduction
- Saturation Process
- Sequent Calculus
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Burel, G. (2014). Cut Admissibility by Saturation. In: Dowek, G. (eds) Rewriting and Typed Lambda Calculi. RTA TLCA 2014 2014. Lecture Notes in Computer Science, vol 8560. Springer, Cham. https://doi.org/10.1007/978-3-319-08918-8_9
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DOI: https://doi.org/10.1007/978-3-319-08918-8_9
Publisher Name: Springer, Cham
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