Formalization of a normalization theorem in simplicial topology

  • Authors:

  • Laureano Lambán;Francisco-Jesús Martín---Mateos;Julio Rubio;José-Luis Ruiz---Reina

  • Affiliations:

  • Department of Mathematics and Computation, University of La Rioja, Logroño, Spain;Computational Logic Group, Department of Computer Science and Artificial Intelligence, University of Seville, Sevilla, Spain s/n. 41012;Department of Mathematics and Computation, University of La Rioja, Logroño, Spain;Computational Logic Group, Department of Computer Science and Artificial Intelligence, University of Seville, Sevilla, Spain s/n. 41012

  • Venue:
  • Annals of Mathematics and Artificial Intelligence
  • Year:
  • 2012

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Abstract

In this paper we present a complete formalization 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. Even if the Normalization Theorem is usually stated as a higher-order result (with a Category Theory flavor) we manage to give a first-order proof of it. To this aim it is instrumental the introduction of an algebraic data structure called simplicial polynomial. As a demonstration of the validity of our techniques we developed a formal proof in the ACL2 theorem prover.