Effective homology of bicomplexes, formalized in Coq

Modelling algebraic structures and morphisms in ACL2

“…Roughly speaking, the representation of a graded structure indexed on a set is achieved thanks to the introduction of an additional parameter, that ranges the elements of the index set, in each operation of the structure. This method differs from the usual approach followed in other ITPs to represent indexed families of structures: a function that takes as input an index, and returns a structure as output (see [20])-this approach is not feasible in the first-order setting of ACL2.…”

“…The Basic Perturbation Lemma [59] was formalised in Isabelle/HOL (see [3]) using the algebraic hierarchy presented in [9]. In the same line, the Effective Homology of the Bicomplexes was formalised in Coq (see [20]) extending the algebraic hierarchy of [25]. These formalisations were related to algorithms and not to the real programs implemented in Kenzo.…”

Computing in Coq with Infinite Algebraic Data Structures

“…In [11] we formalized in Coq the effective homology of a bicomplex. That result can be considered as a generalization of the previous theorem to an infinite (indexed by the natural numbers) family of reductions.…”

“…This formalization was part of the implementation in Coq of the algorithm computing the effective homology of a bicomplex (see [11]). Now, we use the computing capabilities of Coq to explore whether some concrete cones are acyclic or not, as a previous step to proving a general property.…”

“…Section 2 contains some preliminaries on algebraic structures, both in Mathematics and in Coq. Section 3 describes the formalization in Coq of the algorithm computing the effective homology of a cone, in a way that slightly generalizes our previous work in [11]. Concrete Coq instances of chain complexes of infinite type are introduced in Section 4.…”