Efficient Verification of Imperative Programs Using Auto2

Abstract

Auto2 is a recently introduced prover for the proof assistant Isabelle. It is designed to be both highly customizable from within Isabelle, and also have a powerful proof search mechanism. In this paper, we apply auto2 to the verification of imperative programs. We describe the setup of auto2 for both stages of the proof process: verification of a functional version of the program, and refining to the imperative version using separation logic. As examples, we verify several data structures, including red-black trees, interval trees, priority queues, and union-find. We also verify several algorithms making use of these data structures. These examples show that our framework is able to verify complex algorithms efficiently and in a modular manner.

A mechanical proof of the C. T. Yang's Theorem related to a property of derived sets in general topology

“…最近几年, 国内外学者用 Isabelle 证明了大量重要的数学定理, 如 2018-2019 年, 国内学者完成的 有 Li [60] 的 Budan-Fourier 定理形式化、Zhan [61,62] 的 Auto2 prover、Xu 等 [63] 的 "通用图灵机器" 的形式化和 Liu 等 [64]…”

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

“…In the case of data structures (and unlike merge sort), it is useful to state Theorem 4 in terms of yet another, abstract assertion, which hides the concrete reference to the data structure. This follows the technique described in [21,Section 5.3]. Theorems 3 and 4 are the final results for external use.…”

“…First, we discuss automation for reasoning about separation logic with time credits. This is an extension of the setup discussed in [21] for reasoning about ordinary separation logic. Here, we focus on the additional setup concerning time credits.…”

“…First, we incorporate the work of Eberl [11] on the Akra-Bazzi theorem to analyze several divide-and-conquer algorithms. Second, we extend the auto2 prover [21] to provide substantial automation in reasoning about separation logic with time credits, as well as deriving asymptotic behavior of functions.…”

“…As the main theorems to be proved are concerned with asymptotic complexity rather than explicit constants, they do not depend on implementation details. In addition, by using an ad-hoc refinement scheme similar to that in [21], the analysis of an imperative program is divided into clearlyseparated parts: proof of functional correctness, analysis of asymptotic behavior of runtime functions, and reasoning about separation logic. Further separation of concerns is used in amortized analysis.…”