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Discovering frequent trends in time series is a critical task in data mining. Recently, order-preserving matching was proposed to find all occurrences of a pattern in a time series, where the pattern is a relative order (regarded as a trend) and an occurrence is a sub-time series whose relative order coincides with the pattern. Inspired by the order-preserving matching, the existing order-preserving pattern (OPP) mining algorithm employs order-preserving matching to calculate the support, which leads to low efficiency. To address this deficiency, this paper proposes an algorithm called efficient frequent OPP miner (EFO-Miner) to find all frequent OPPs. EFO-Miner is composed of four parts: a pattern fusion strategy to generate candidate patterns, a matching process for the results of sub-patterns to calculate the support of super-patterns, a screening strategy to dynamically reduce the size of prefix and suffix arrays, and a pruning strategy to further dynamically prune candidate patterns. Moreover, this paper explores the order-preserving rule (OPR) mining and proposes an algorithm called OPR-Miner to discover strong rules from all frequent OPPs using EFO-Miner. Experimental results verify that OPR-Miner gives better performance than other competitive algorithms. More importantly, clustering and classification experiments further validate that OPR-Miner achieves good performance.