Refinements for Free!

Abstract

International audienceFormal verification of algorithms often requires a choice be-tween definitions that are easy to reason about and definitions that are computationally efficient. One way to reconcile both consists in adopt-ing a high-level view when proving correctness and then refining stepwise down to an efficient low-level implementation. Some refinement steps are interesting, in the sense that they improve the algorithms involved, while others only express a switch from data representations geared towards proofs to more efficient ones geared towards computations. We relieve the user of these tedious refinements by introducing a framework where correctness is established in a proof-oriented context and automatically transported to computation-oriented data structures. Our design is gen-eral enough to encompass a variety of mathematical objects, such as rational numbers, polynomials and matrices over refinable structures. Moreover, the rich formalism of the Coq proof assistant enables us to develop this within Coq, without having to maintain an external tool

Formally Verified Certificate Checkers for Hardest-to-Round Computation

“…For the moment, this is done in an ad-hoc way using morphisms between data-structures and explicitly applying rewriting rules for the correctness proof. The use of a more systematic approach such as the ones proposed in [10,19,23] would be a clear improvement.…”

“…Let us denote by the final value of k (at the end of the while loop). Then notice that the successive values of variables (a, b) in this loop correspond to a sequence (a k , b k ) 0 k that satisfies relations (9) and (10). Given that (7) holds by hypothesis, and that (8) is trivially satisfied by the integral root (x, y), we can apply Lemma 2.…”

QED at Large: A Survey of Engineering of Formally Verified Software

“…Since then, there have been a number of refinement tools in Isabelle/HOL with support for logic (Hemer et al, 2001), object-oriented (Liu et al, 2011), functional (Lammich, 2013), and imperative (Lammich, 2015) programs. Cohen et al (2013) developed a framework for Coq called CoqEAL which automates key steps of data refinement. Delaware et al (2015) presented Fiat, a refinement framework for deductive synthesis of abstract data types in Coq.…”

Proof–Based Synthesis of Sorting Algorithms for Trees

“…The generated algorithms can be checked for conformity with the input specification by validating the proof trails for each refinement process, for example using the Coq library Fiat [7] to ensure the soundness of the validation step by certification with the Coq kernel. [6] presents a different Coq library using datatype refinement to verify parameterized algorithms for which the soundness proof of some version can be deduced from that of a previous (less efficiently implemented) version. Implementing inference rules directly in Coq may be of interest if one can prove that every generated synthesized algorithm is sound.…”