What is meant by the Mushaf of Imam Ali (as) is the Quran that he compiled after the passing of the prophet (pbuh). This Quran has certain traits that other Qurans dont, namely:
1- The exact order of the verses and surahs, according to the chronological order of their revelation. In other words, in this Quran, all Makki verses precede Madani ones and the order of their revelation is clear, showing the development of Islamic rulings over the 23 years of its revelation and which verses are abrogating and which ones are abrogated.
2- In this Quran, the original recitation of the prophet (pbuh) has been used, thus leaving no questions regarding the recitation of some of the verses as happened later in history. This would lead to the precise understanding and interpretation of the verses, making it a very significant characteristic that this Quran had. Difference in recitation sometimes might be misleading to the person interpreting the verse, something that this Quran was devoid of.
3- This Quran comprised of the occasions of revelation of verses, meaning that it had explained all the incidents that had caused the revelation of certain verses and surahs in its margins. These margins would have been the best means of understanding the verses and getting the answers to many questions. In addition to this, there were also tawils which were general and comprehensive conclusions drawn from specific incidents mentioned in the verses which would have also greatly helped in understanding the verses. The imam himself says: I put together a book for them that contained tawil and tanzil (occasions of revelation).
Based on the sources of both Islamic schools of thought, this book certainly exists, and as a matter of fact, the Sunni sources on this subject surpass Shia ones in number.
Among Shia sources, this Quran has been mentioned in the book of Suleym ibn Qeys, in Kafi by Abu Jafar Kuleini (328 ah), Kitabul-Tafsir, also known as Tafsir of Ayyashi, by Masoud Ayyashi (of the Islamic scholars of the fourth century ah), Al-Itiqadat of Abu Jafar Saduq (386 ah), and Manqib Ale Abi Taleb by Ibn Shahr Ashub Mazandarani (588 ah). Ibn Shahr Ashub has engaged in this subject by listing several Sunni sources. It can be said that all of these books report the compilation of this Quran in the same way; that after the demise of the prophet (pbuh), Imam Ali (as) sat at home and completed its compilation.
As for Sunni sources that have mentioned this Quran up until the eighth century, they are: Al-Tabaqatul-Kubra by Muhammad ibn Sad (230 ah), Fadailul-Quran by Ibn Dhurays (294 ah), Kitabul-Masahif by Ibn Abi Dawud (316 ah), Kitabul-Fihrist by Ibn Nadim who reports from Ahmad ibn Jafar Munada, better known as Ibn Munada and Al-Masahif by Ibn Ashtah (360 ah), Hilyatul-Owliya wa Tabaqatul-Asfiya and Al-Arbain by Abi Naim Al-Isfihani (430 ah), Al-Istiab fi Marifatil-Ashab by Ibn Abdil-Birr (463 ah) which has narrated through two ways, Shawahidul-Tanzil by Hakem Haskani (a fifth century scholar) who has proven the existence of this book through numerous proofs, Mafatihul-Asrar wa Masabihul-Anwar by Abdul-Karim Shahrestani (548 ah) (amongst all Sunni scholars, this scholar has reported this Quran in detail, explaining how it was compiled, how the imam offered it to the Sahabah and how they refused to accept it and how he brought them reasoning and arguments in order to persuade them), Al-Manqib by Khatib Kharazm (568 ah), Al-Tashil fi Ulumil-Tanzil by Ibn Jizy Kalbi (741 ah).
Ibn Abi Dawud has reported this Quran through Ashath ibn Sawwar and says: No one other than Ashath has any narration about this Quran. This statement clearly shows that whoever said it is uninformed about the different chains of narrators that back this issue. For instance, Ibn Dhurays narrates it through a different chain of narrators, Ibn Hajar Asqalani and Mahmoud ibn Ahmad Ayni also consider it a weak one because of a problem (inqita) in the chain of narrators, nevertheless, this problem can also be seen in other chains of narrators. There are other chains of narrators that report this Quran that are authentic, like the one related to Ibn Dhurays who narrates from Muhammad ibn Sirin who narrates from Akramah who narrated from Imam Ali (as).
Alusi Baghdadi (1270 ah) has also made a statement regarding this matter. He believes that some of the chains of narrators to be Mowdhu (موضوع) and some to be Dhaif (ضعیف) and only knows Ibn Dhurayss narration to be authentic. In that narration Muammad ibn Sirin asks Akramah: Did [Imam] Ali arrange the verses according to the order of their revelation in that Quran? Despite this clarity, Alusi says what is meant by this narration is the memorization of the Quran [by Imam Ali (as), and not its compilation], so it is said.
What Alusi is probably saying is that the imam memorized the Quran in the order its verses were revealed!!
In addition to the strange statement above, Alusi and others havent evaluated the chains of narrators of these hadiths in a scholarly way, because hadiths that have the same content as an authentic hadith are considered mutabi and shaheds in the science of hadiths (Dirayah) and arent looked at as completely worthless; hadiths that Alusi thinks of as dhaif and modhu. Since they have the same content as an authentic hadith, they actually back and strengthen the authentic one, granting it istifadhah (which makes a hadith more authentic). Therefore, evaluating each hadith by itself without looking at all of them in their entirety is something unscholarly and not in correspondence with the principles of the science of hadiths.
Another point that must be made here is that the term جمع القرآن (compilation of Quran) which has been mentioned in some of these hadiths cant mean the memorization of the Quran (although it might mean this in other instances of usage), because as our Sunni brothers themselves say, after the khalifah was chosen after the demise of the prophet (pbuh), Imam Ali (as) had nothing to do except stay at home and memorize the Quran. Some Sunni scholars see this as a justification for the endless knowledge of the imam compared to the other khalifahs saying: Most of the interpretations of the Quran that have been left behind by the khalifahs belong to [Imam] Ali because he had been put aside from the caliphate until after Uthman.
Although this is a very unsupported and baseless thing to say, but the question remains that if the imam had nothing to do, then what was the need for him to stay at home and swear that he wont wear his cloak and?! Also, according to many Sunni hadiths (that are so much in number that they leave no room for any doubt, making them mutawatir in hadithic terms) regarding the verse لِنَجْعَلَها لَکُمْ تَذْکِرَهً وَ تَعِیَها أذُنٌ واعِیَهٌ, the prophet (pbuh) told [Imam] Ali (may Allah be pleased with him): I have asked Allah (swt) to make you who this verse is speaking of., and it is because of this Imam Ali (as) would say: Whatever I hear, I never forgot and never will forget.
It is as a result of this that not only did Imam Ali (as) have the Quran memorized, but had all the knowledge the prophet (pbuh) had endowed him with in his heart. According to some hadiths, the Quran was compiled all at once by the prophets decree and to uphold the thaqalain and clearly, no one deserved to do such more than Imam Ali (as). Even if we assume that the hadith of AbduKheyr that quotes Imam Ali (as) saying that Abu Bakr was the first to put the Quran together has an acceptable chain of narrators, it is in conflict with another hadith that says the first to do so was Imam Ali (as), and because the previous is a khabar wahed (خبر واحد) while this hadith is a mustafidh (مستفیض) one, the khabar wahed becomes a (شاذ) hadith.
Therefore, according to the sources of both Islamic schools of thought, the existence of the the Mushaf of Imam Ali (as) is for sure and there is no room for any doubt and question regarding the proofs for its existence. For further information, see the following books:
1- Mushafe Ali (as) va she Mushafe Digar, Dr. Jafar Nekounam
2- Pajouheshi dar Mushafe Imam Ali (as) va She Mushafe Digar, Dr. Jafar Nekounam.
 Mohammad Javad Balaghi, Alaul-Rahman, vol. 1, pg. 257 و لقد جئتهم بالکتاب مشتملا علی التنزیل و التاویل From the website: Andisheye Qom
 Suleym ibn Qeys, previous, pp. 581-582, 660 and 665.
 Abu Jafar Kuleini, Al-Kafi, the book of the virtues of the Quran (Fadailul-Quran), vol. 2, pg. 633, hadith 29.
 Muhammad ibn Masoud Ayyashi, Tafsir Ayyashi, vol. 2, pg. 307.
 Abu Jafar Saduq, Al-Itiqadat, pg. 81.
 Muhammad ibn Ali ibn Shahr Ashub, Manaqib Ale Abi Taleb, vol. 2, pp. 50-51.
 Ibid, vol. 2, pg. 52.
 Muhammad ibn Sad, Al-Tabaqatul-Kubra, vol. 2, pg. 328.
 Muhammad ibn Ayyub ibn Dhurays, Fadhailul-Quran (research of Urwah Badir), pg. 36.
 Suleyman ibn Ashath, Kitabul-Masahif (revised by Atur Jafri, pg. 61).
 Muhammad ibn Ishaq, Kitabul-Fihrist, pg. 31-32.
 Quoted by: Jalaluddin Suyuti, Al-Itqan, vol. 1, pg. 58.
 Abu Naim Isfihani, Hilyatul-Owliya wa Tabaqatul-Asfiya, vol.1, pg. 67.
 Quoted from: Ibn Shahr Ashub, ibid, vol. 2, pg. 50.
 Yusuf ibn Abdil-Birr, Al-Istiab fi Marifatil-Ashab (researched by Muhammad Ali Bajawi), third section, pg. 974.
 Hakem Haskani, Shawahidul-Tanzil li Qawaidil-Tafdhil (researched by Muhammad Baqir Mahmoudi), vol. 1, pp. 36-38.
 Abdul-Karim Shahrestani, Mafatihul-Asrar wa Masabihul-Anwar, vol. 1, pg. 121.
 Muwaffaq ibn Ahmad Kharazmi, Al-Manaqib (researched by Muhammad ibn Malek Mahmoudi), pg. 94.
 Ulumul-Quran indal-Mufassirin, quoted by: Markazul-Thiqafah wal-Maariful-Quraniyyah, vol. 1, pg. 351.
 Suleiman ibn Ashath, Kitabul-Masahif, pg. 16.
 Muhammad ibn Ayyub ibn Dhurays, ibid, pg. 36.
 Ibn Hajar Asqlani, ibid, vol. 9, pp. 12-13, and also see: Mahmou ibn Ahmad Ayni, Umdatul-Qari (commentary of Sahih Bukhari).
 Muhammad ibn Ayyub ibn Dhurays, ibid, pg. 36.
 Narrated through AbuHayyan Tawhidi. Mahmoud Alusi, Ruhul-Maani, vol. 1, pg. 41.
 Narrated through Ibn Sirin. Ibid.
 Ibid, vol. 1, pg. 41.
 Mutabi and Shahed hadiths are important because they are verification for the content of other hadiths, thus the authenticity of their chain of narrators isnt paid attention to. See: Jalaluddin Suyuti, Tadribul-Rawi fi Sharhi Taqribil-Nawawi, pg. 202.
 Jalaluddin Suyuti, Al-Itqan, vol. 2, pg. 187, and Ibn Atiyyah Andalusi, research of Muhammad Abdul-Salam Abdul-Shafi, vol. 1, pg. 13.
 Because first of all, During his reign, Imam Ali (as) has presented the interpretation of verses and explanation of traditions from the prophet (pbuh) especially in his sermons, and secondly, Uthman ibn Affan, the third khalifah who was far from the khilafah during the reign of the first and second khalifahs, is the only one who has hadiths on tafsir of Quranic verses. See: Alauddin Ali Al-Muttaqi Al-Hindi, Kanzul-Ummal fi Sunanil-Aqwal wal-Afal, chapter on tafsir, vol. 2, pg. 353.
It should also be noted that the hadiths on the tafsir of Quranic verses in Kanzul-Ummal that have come from the six Sihah total to 544 in which 290 belong to Imam Ali (as) and the rest to the other Sahabah. See: Ali Muttaqi Hindi, ibid.
 Tabari, Jameul-Bayan fi Tawilil-Quran (also known as Tafsir Tabari), vol. 29, pg. 31. For further information on the proofs for this hadith, see: Ahmad ibn Yahya Al-Baladheri, Ansabul-Ashraf (researched by Sheikh Muhammad Baqir Mahmoudi), vol. 1, pg. 34. انّ النبی(ص) قال لعلی- کرم الله وجهه- إنّی دعوت الله تعالی أن یجعلها أذنک یا علی و قال علی کرم الله وجهه: فما سمعت شیئاً فنسیته و ما کان لی أن أنسی
 Because Imam Ali (as) is the gate of the city of knowledge of the prophet (pbuh) (referring to the famous hadith that says the prophet is the city of knowledge and Ali is its gate) and no one in the past and future has ever surpassed and ever will surpass him in knowledge, and as he says: By Allah, no verse was revealed, except that I know about what, where and regarding whom it was revealed. The late Seyyid Abdul-Husein Shafaruddin in his invaluable book of Al-Murajiat has cited these hadiths, and the researcher of the book, Sheikh Husein Al-Radhi, has listed the chains of narrators of each hadith in his Al-Hawamishul-Tahqiqiyyah that is attached to the book. See: Husein Al-Radhi, Al-Hawamishul-Tahqiqiyyah attached to the book of Al-Murajiat, pp. 425-431.
 Ibn Abi Dawud, Kitabul-Masahif, vol. 2, pg. 338.
 Although Ibn Abi Dawud narrates this hadith through five different chains of narrators, but since all the chains end in Sufyan narrating from Sady from Abd Kheyr from Imam Ali (as), this hadith is technically a khabar wahed.
 A Shadh hadith is a hadith with an acceptable chain of narrators that is in conflict with a hadith that has more credibility than it. This hadith has another problem too, and that is that some of what is narrated in it doesnt read with what has been recorded in history, because before the compilation of the Quran by Zayd ibn Thabit, others such as Ibn Masoud, Ubayy ibn Kab, etc. had their own mushafs.
 For more information on the proofs for the Mushaf of Imam Ali (as), see: Maqalat va Barresiha Magazine (Theology Dpt. Of Tehran University, Article: Barresi va Naqde Asnade Mushafe Imam Ali (as) dar Manabee Fariqayn), 68th book, pp. 23-45.