What is the evidence that Ali (peace be upon him) did not lead the prayer after the Prophet Muhammad (peace and blessings be upon him)?

It is among the fundamentals of Ahl al-Sunnah (definition) wa al-Jama‘ah that their hearts and tongues are sound regarding the Companions of the Messenger of Allah (peace be upon him). They accept what has been transmitted about their virtues and ranks, and they give precedence to those who spent and fought before the Conquest [of Makkah], and to the Muhajirun over the Ansar. They believe in Allah's forgiveness for the people of Badr and that none of those who pledged allegiance under the tree will enter the Fire. They affirm that the best of the Ummah after its Prophet are Abu Bakr, then Umar, then Uthman, then Ali. They bear witness to Paradise for those for whom the Messenger of Allah (peace be upon him) bore witness.

The one most entitled to the Caliphate after the Messenger of Allah (peace be upon him) is Abu Bakr al-Siddiq. Several hadiths indicate his leadership, among them the Prophet's command (peace be upon him) to return to Abu Bakr if he was not found, the command to follow the example of Abu Bakr and Umar, the Prophet's vision (peace be upon him) which was interpreted as the duration of Abu Bakr and Umar's rule, and the Prophet's appointment (peace be upon him) of Abu Bakr as Imam for the Muslims in prayer during his illness before his death, which was considered an indication of his succession in the greater Imamate.