The best replacement for petition is motion.

The best alternatives to petition are rated as following:
★★★★★  motion
★★★★  complaint
★  application
★  request
  pleading
  supplication
  intercession
  post petition
  initiative
  proposal
  appeal

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Speaking about petition, one probably shouldn't forget about united states tax court.

Other search terms related to petition and their ratings are:
★★★★★ united states tax court
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