The proof development process in interactive theorem provers (ITPs) requires the users to manually search for proofs by interacting with proof assistants. The activity of finding the correct proofs can become quite cumbersome and time consuming for users. To make the proof searching process easier in proof assistants, we provide an evolutionary/heuristic-based framework. The basic idea for the framework is to first generate random proof sequences from a population of frequently occurring proof steps that are discovered with sequential pattern mining. Generated proof sequences are then evolved till their fitness match the fitness of the target (or original) proof sequences. Three algorithms based on the proposed framework are developed using the Genetic Algorithm (GA), Simulated Annealing (SA) and Particle Swarm Optimization (PSO). Extensive experiments are performed to investigate the performance of the proposed algorithms using the HOL4 proof assistant. Results have shown that the proposed algorithms can efficiently evolve the random sequences to obtain the target sequences. In comparison, PSO performed better than SA and SA performed better than GA. In general, the experimental results suggest that combining evolutionary/heuristic algorithms with proof assistants allow efficient support for proof finding/optimization.

交互式定理证明者（ITP）中的证明开发过程要求用户通过与证明助手进行交互来手动搜索证明。对于用户而言，寻找正确证明的活动可能变得相当繁琐且耗时。为了简化证明助手中的证明搜索过程，我们提供了一个基于进化/启发式的框架。该框架的基本思想是首先从通过顺序模式挖掘发现的一系列频繁发生的证明步骤中生成随机证明序列。然后进化生成的证明序列，直到它们的适合度与目标（或原始）证明序列的适合度匹配为止。使用遗传算法（GA），模拟退火（SA）和粒子群优化（PSO）开发了基于所提出框架的三种算法。使用HOL4证明助手进行了广泛的实验，以研究所提出算法的性能。结果表明，所提出的算法可以有效地进化随机序列以获得目标序列。相比之下，PSO的性能优于SA，SA的性能优于GA。总的来说，实验结果表明，结合进化/启发式算法和证明助手可以为证明查找/优化提供有效的支持。