Does the prostration of forgetfulness (sujood al-sahw) before the salutation include forgetting the takbir for bowing and the tashahhud immediately after waking up from sleep?

The takbir (pronouncement of "Allahu Akbar") during bowing is obligatory. Doubting its omission necessitates performing the prostration of forgetfulness (sujud al-sahw) before the taslim (salutation). Two prostrations of forgetfulness suffice for omitting one or more obligations. However, doubting the omission of a pillar, such as the final tashahhud in the Fajr prayer, is not compensated by sujud al-sahw; rather, it must be performed. If it is not performed, the prayer becomes invalid and must be repeated, because the default assumption is that it was not performed. Doubt concerning the omission of a pillar is not considered if it is a mere illusion, or frequent as is the case with those afflicted by waswas (obsessive whispering), or after finishing the prayer. However, genuine doubt during the prayer is considered as though the pillar was omitted, and it must be performed. If one predominantly believes that he has performed it, then he relies on this belief and performs sujud al-sahw after the taslim.