The past participle of “do” is “done,” so it’s not “they have did what they promised not to do” but “they have done. . . .” But without a helping verb, the word is “did.” Nonstandard: “I done good on the test.” Standard: “I did well on the test.” Using “done” itself as a helping verb is also a nonstandard dialectal pattern: “he done give us Christmas tree“ in standard English would be “he gave us a Christmas tree.”