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High utility sequential pattern mining (HUSPM) is a significant and valuable activity in knowledge discovery and data analytics with many real-world applications. In some cases, HUSPM can not provide an excellent measure to predict what will happen. High utility sequential rule mining (HUSRM) discovers high utility and high confidence sequential rules, allowing it to solve the problem in HUSPM. All existing HUSRM algorithms aim to find high-utility partially-ordered sequential rules (HUSRs), which are not consistent with reality and may generate fake HUSRs. Therefore, in this paper, we formulate the problem of high utility totally-ordered sequential rule mining and propose two novel algorithms, called TotalSR and TotalSR+, which aim to identify all high utility totally-ordered sequential rules (HTSRs). TotalSR creates a utility table that can efficiently calculate antecedent support and a utility prefix sum list that can compute the remaining utility in O(1) time for a sequence. We also introduce a left-first expansion strategy that can utilize the anti-monotonic property to use a confidence pruning strategy. TotalSR can also drastically reduce the search space with the help of utility upper bounds pruning strategies, avoiding much more meaningless computation. In addition, TotalSR+ uses an auxiliary antecedent record table to more efficiently discover HTSRs. Finally, there are numerous experimental results on both real and synthetic datasets demonstrating that TotalSR is significantly more efficient than algorithms with fewer pruning strategies, and TotalSR+ is significantly more efficient than TotalSR in terms of running time and scalability.