Effective homology of bicomplexes, formalized in Coq

In this paper, we present a complete formalization in the Coq theorem prover of an important algorithm in computational algebra, namely the calculation of the effective homology of a bicomplex. As a necessary tool, we encode a hierarchy of algebraic structures in constructive type theory, including graded and infinite data structures. The experience shows how some limitations of the Coq proof assistant to deal with this kind of algebraic data can be overcome by applying a separation of concerns principle; more concretely, we propose to distinguish in the representation of an algebraic structure (such as a group or a module) a behavioural part, containing operation signatures and axioms, and a structural part determining if the algebraic data is free, of finite type and so on.