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We present a complete mechanized proof of the result in homological algebra known as basic perturbation lemma. The proof has been carried out in the proof assistant Isabelle, more concretely, in the implementation of higher-order logic (HOL) available in the system. We report on the difficulties found when dealing with abstract algebra in HOL, and also on the ongoing stages of our project to give a certified version of some of the algorithms present in the Kenzo symbolic computation system.

In this paper, a set of programs enhancing the Kenzo system is presented. Kenzo is a Common Lisp program designed for computing in Algebraic Topology, in particular it allows the user to calculate homology and homotopy groups of complicated spaces. The new programs presented here entirely compute Serre and Eilenberg-Moore spectral sequences, in particular the E r p,q and d r p,q for arbitrary r. They also determine when E r p,q = E ∞ p,q and describe the filtration of the target homology groups H p+q by the E ∞ p,q 's.

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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.

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When the results of a computer program are compared to some theorems proved on a theoretical basis three situations can occur: there can be an agreement between both approaches, the computer program can obtain calculations not covered by the theorems, or a discrepancy can be found between both methods. In this paper we report on a work where the three above mentioned situations happen. We have enhanced the Computer Algebra called Kenzo to deal with the computation of homotopy groups of suspended classifying spaces, a problem tackled by Mikhailov and Wu in a paper published in the journal Algebraic and Geometric Topology. Our experimental approach, based on completely different methods from those by Mikhailov and Wu, has allowed us in particular to detect an error in one of their published theorems.