Abstract
In this paper we present a complete formalization, using the ACL2 theorem prover, of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. The interest of this work stems from three sources. First, the normalization theorem is the basis for some design decisions in the Kenzo computer algebra system, a program for computing in Algebraic Topology. Second, our proof of the theorem is new and shows the correctness of some formulas found experimentally, giving explicit expressions for the above-mentioned homotopy equivalence. And third, it demonstrates that the ACL2 theorem prover can be effectively used to formalize mathematics, even in areas where higher-order tools could be thought to be more appropriate.
Partially supported by Ministerio de Ciencia e Innovación, project MTM2009-13842, and by European Commission FP7, STREP project ForMath.
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Lambán, L., Martín-Mateos, F.J., Rubio, J., Ruiz-Reina, J.L. (2011). Applying ACL2 to the Formalization of Algebraic Topology: Simplicial Polynomials. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds) Interactive Theorem Proving. ITP 2011. Lecture Notes in Computer Science, vol 6898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22863-6_16
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DOI: https://doi.org/10.1007/978-3-642-22863-6_16
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