Is it permissible to change one's intention after entering into prayer upon remembering that one did not intend a specific prayer?

If a worshipper doubts whether he intended to pray or not, or if he intended to pray Dhuhr or Asr, and he remembers his intention before doing anything else while in doubt, then he may proceed with his prayer and it is valid. However, if he does something while in doubt, his prayer becomes invalid according to many scholars, although some do not consider it invalid. If he remembers the specific intention before starting another pillar, his prayer is valid. But if he starts another pillar before remembering the specific intention, his prayer is invalid and he must repeat it. As for making a mistake in the intention, such as intending to pray Isha while the prayer he is performing is Maghrib, his prayer is invalid because it is not permissible to change the intention from one obligatory prayer to another during the prayer.