Abstract
We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. These axioms have the property of being easily orientable into rewrite rules. This allows us to give finite first-order axiomatizations of arithmetic and real fields theory, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework.
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Kirchner, F. (2007). A Finite First-Order Theory of Classes. In: Altenkirch, T., McBride, C. (eds) Types for Proofs and Programs. TYPES 2006. Lecture Notes in Computer Science, vol 4502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74464-1_13
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DOI: https://doi.org/10.1007/978-3-540-74464-1_13
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