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

Abstract

International audienceWe apply current theorem proving technology to certified code in the domain of abstract algebra. More concretely, based on a formal proof of the (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

Computing in Coq with Infinite Algebraic Data Structures

“…This paper continues our previous work in translating Sergeraert's ideas to theorem provers [2][3][4]1], with the aim of formalizing this part of algorithmic mathematics and, more importantly, of applying formal methods to the study of the Kenzo system [12] (a Common Lisp program developed by Sergeraert to implement effective homology algorithms). The first important milestone in this area was the mechanized proof in the Isabelle/HOL proof assistant of the Basic Perturbation Lemma (BPL), published in [2].…”

“…This formal proof was carried out in the Higher Order Logic (HOL) built on top of Isabelle, and therefore extracting programs from it was not a simple task. The findings on this topic were reported in [3]. A different approach is being carried out by T. Coquand and A. Spiwack [9] who are using Coq to model a part of Category Theory, and then trying to obtain a BPL proof in this larger context.…”

Formalization of a normalization theorem in simplicial topology

“…This is the reason why a project to apply formal methods to the study of Kenzo as a software system was launched some years ago [6,12]. Eventually, this research line arrived to the formalization of some parts of Algebraic Topology and Homological Algebra by using proof assistants as Isabelle/HOL [2,3] or Coq [7]. A different approach to using Coq to implement in constructive type theory some features of Kenzo can be found in [4].…”

Modelling algebraic structures and morphisms in ACL2

“…These formalisations were related to algorithms and not to the real programs implemented in Kenzo. The problem of extracting programs from the Isabelle/HOL proofs was studied [4], but even there, the programs are generated in ML, far from Kenzo.…”