Generating certified code from formal proofs: a case study in homological algebra

We apply current theorem proving technology to certified code in the domain of abstract algebra. More concretely, based on a formal proof of the Basic Perturbation Lemma (a central result in homological algebra) in the prover Isabelle/HOL, we apply various code generation techniques, which lead to certified implementations of the associated algorithm in ML. In the formal proof, algebraic structures occurring in the Basic Perturbation Lemma are represented in a way, which is not directly amenable to code generation with the available tools. Interestingly, this representation is required in the proof, while for the algorithm simpler data structures are sufficient. Our approach is to establish a link between the non-executable setting of the proof and the executable representation in the algorithm, which is to be generated. This correspondence is established within the logical framework of Isabelle/HOL—that is, it is formally proved correct. The generated code is applied to and illustrated with a number of examples.