Formalization of a normalization theorem in simplicial topology

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. This work is dedicated to our colleague and friend Mirian Andrés. She started this research but passed away at the age of only 29 due to a car accident. Mirian, the best friend for your friends, we do not forget you.

Modelling algebraic structures and morphisms in ACL2

“…We are mainly interested in the Kenzo system, where the methodologies and tools presented in this paper can reduce the formalisation effort in works like [44,47].…”

“…loops were replaced by tail-recursive functions). The work presented in [2,35,36,50] did not concern algebraic structures, but ACL2 has also been used to formalise constructions (the Normalisation theorem and the Eilenberg-Zilberg theorem [59]) involving algebraic structures implemented in Kenzo, see [44,47]. In contrast to the work in Isabelle and Coq, the algebraic structures involved in the ACL2 formalisations were developed from scratch instead of using, as a basis, a previously developed algebraic hierarchy.…”

“…The formalisations presented in [44,47] show that it is possible to reason about algebraic concepts in ACL2; however, the first-order quantifier-free logic of ACL2 stands in the way of a widespread use of this ITP to work with algebraic structures. In this paper, we show how this weakness can be overcome thanks to two of ACL2's great strengths: programmable extensibility and proof automation.…”

“…In the next section, we present a brief introduction to the ACL2 system and the tools employed in our development. Section 3 presents how algebraic structures can be modelled from scratch in ACL2 (this is the approach followed in [44,47]), and the difficulties that are associated with this process. To overcome those problems, we present a methodology (by means of examples) to implement a set of tools for algebraic structures in Sect.…”

Certified symbolic manipulation

“…In this area of ACL2 applications to Algebraic and Simplicial Topology, several contributions have been already made [1, 7,9]. Our last development is a complete ACL2 proof of the so-called Eilenberg-Zilber theorem [8].…”