1) The knowledge of the Imam's (a.s) is what they inherit from Prophet Muhammad (saww):

It should be known that inheritance of knowledge is not like inheriting a material thing, therefore inheritance of knowledge does not necessarily mean that the Imam's (a.s) did not already know that. We have many narrations which prove that Imam Ali (a.s) had lot of knowledge even during the life of Prophet Muhammad (saww). Hence this rules out knowledge by inheritance and gives rise to the second source of Imam's (a.s) knowledge.

Inheriting the knowledge from the Holy Prophet (saww) mainly indicates that the Imam's (a.s) are the true successors of the Prophet (saww) and not that they did not know.

We have the following authentic narration which says the the true 'Knowledge' is above and beyond what is inherited by the Imam's (as) from Prophet Muhammad (saww):

حدثنا محمد بن عبد الحميد عن يونس بن يعقوب عن منصور بن حازم عن ابى عبد الله عليه السلام قال قلت ان الناس يذآرون ان عندآم صحيفة طولها سبعون ذراعا فيها ما يحتاج إليهالناس وان هذا هو العلم فقال أبو عبد الله عليه السلام ليس هذا هو العلم انما هو اثر عن رسول الله ان العلم الذى يحدث في آل يوم وليلة

It has been narrated to us Muhammad Bin Abdul Hameed, from Yunus Bin Yaqoub, fromMansour Bin Haazim, who has said:

I said to Abu Abdullah (as), "The people mention that with You is a Parchment the length of seventy cubits in which is all that is needed for the people, and this is the knowledge."

Abu Abdullah (as) said, ‘This is not the knowledge, but these are the effects from the Messenger of Allah (saww). The knowledge is what takes place every day and night."

[Source: Basaair Al-Darajaat, Vol. 3, Chapter. 11, Hadees. 3]

Narration is Authentic/Saheeh as per the chain of narrators. There are multiple narrations with the same meaning.

2) The knowledge of the Imam's (a.s) as taught/provided/given by Prophet Muhammad (saww):

Again the way we understand this is that the passing/teaching of knowledge from Prophet Muhammad (saww) to the Imam's (as) or from one Imam (a.s) to another is not like what we commonly have amongst ourselves. Simply because if we read narrations which describe the extent of the knowledge of the Imam's (a.s), it would be impossible for it to be taught in the manner a teacher teaches a student. So here the passing by of knowledge is not bound by time or any other constraint. Also it will not be possible to limit as to what extent and when the knowledge was passed from one Imam (a.s) to another.

For example, we have a narration with authentic chain of narrators in which Imam Baqir (a.s) says,

"Do you not consider that how would Allah, the Most Holy, the Most High, make it obligatory to obey those who possess Divine authority over his servants and then hide from them (a.s) `the news of the heaves and the earth? How would He cut them off of the sources of knowledge that might come to them to maintain their religion?"

Read the complete narration here: http://www.marefateahlebait.com/know-the-ahlul-bait/knowledge-of-the-imam

Another narration with an authentic chain which says that Imam Hasan (a.s) knows 70,000 languages. Read the narration here:

http://www.marefateahlebait.com/know-the-ahlul-bait/authority-of-imam-hasan-as

We also have many narrations which say that the Imam's (as) say somethings when they come on the earth and give the Testimonies. So who taught this to the Imam's (a.s)??

So the question comes through whom and when was this knowledge given to the Imam's (a.s)??

This gives us another source of the knowledge of the Imam's (a.s)

3) Knowledge provided by Allah (swt):

Here we have many narrations which say that the AhlulBait (as) have knowledge provided by Allah (swt) from their very creation and from their very birth.

Narrated from Imam Jafar As-Sadiq (as):

[Quoting only a part of the narration here]

وإذا وقع من بطن امه وقع واضعا يديه على الارض رافعا رأسه إلى السماء فأما وضعه يديه على الارض فإنه يقبض كل علم لله أنزله من السماء إلى الارض وأما رفعه رأسه إلى السماء فان مناديا ينادي به من بطنان العرش من قبل رب العزة من الافق الاعلى باسمه واسم أبيه يقول: يا فلان بن فلان اثبت تثبت، فلعظيم ما خلقتك أنت صفوتي من خلقي وموضع سري وعيبة علمي وأميني على وحيي وخليفتي في أرضي، لك ولمن تولاك أوجبت رحمتي ومنحت جناني وأحللت جواري، ثم وعزتي وجلالي لاصلين من عاداك أشد عذابي وإن وسعت عليه في دنياي من سعة رزقي فإذا انقضى الصوت - صوت المنادي - أجابه هو واضعا يديه رافعا رأسه إلى السماء يقول " شهد الله أنه لا إله إلا هو والملائكة واولوا العلم قائما بالقسط لا إله إلا هو العزيز الحكيم " قال: فإذا قال ذلك أعطاه الله العلم الاول والعلم الآخر واستحق زيارة الروح في ليلة القد

When he comes out of his mothers womb he places his hands on the floor and raises his head to the sky. Placing his hands on earth is an indication of his taking possession of all the knowledge of Allah that is sent from the heavens to earth. Raising his head to the sky is that because a caller from inside the Throne calls him from the high horizon with his name and the name of his father. It is for (on behalf of) the Lord, the Most Majestic, the Most Gracious, saying, "O so and so, son of so and so, be firm so that you would be established. For the great purpose that I have created you, you are my chosen one in my creatures. You are the keeper of My secrets, the container of My knowledge, the trustee of My inspiration and My deputy and representative on earth. I have made it necessary for My blessings and mercy to reach you and those who would love you (and acknowledge your Divine Authority). I have gifted My paradise and given a place near Myself to you and to them. Also, by My Majesty and Grace, I will make your enemies to feel the heat of My punishment even though I may have given them expanded prosperity in My world such as prosperity in their livelihood. When the call of caller ends he responds to it by placing his hands on the ground and raising his head to the sky and says the following.

"Allah Himself testifies that He is the only Lord. The angels and the men of knowledge and justice testify that Allah is the only Lord, the Majestic, and All-wise." (3:18)

The Imam said, "When he says this, Allah grants him the knowledge of the first and the knowledge of the last the qualification to be visited by the spirit in the nights of destiny."

[Source: Al-Kafi, Kitab Al-Hujja, Chapter: 93, Hadees. 1]

حدثنا احمد بن محمد عن الحسن بن محبوب عن صالح بن سهل الهمداني و غيره رواه عن يونس بن ظبيان عن ابى عبد الله عليه السلام قال إذا اراد الله ان يقبض روح امام و يخلقمن بعده اماما انزل قطرة من ماء تحت العرش إلى الارض فيلقيها على ثمرة أو على بقلة فيأآل تلك الثمرة أو تلك البقلة الامام الذى يخلق الله منه نطفة الامام الذى يقوم من بعده قالفيخلق الله من تلك القطرة نطفة في الصلب ثم يصير إلى الرحم فيمكث فيها اربعين ليلة فإذا مضى له اربعون ليلة سمع الصوت فإذا مضى له اربعة اشهر آتب على عضده الايمن وتمتآلمة ربك صدقا وعدلا لا مبدل لكلماته وهو السميع العليم فإذا خرج إلى الارض اوتى الحكمة وزين بالعلم والوقار والبس الهيبة وجعل له مصباح من نور يعرف به الضمير ويرى بهاعمال العباد

It has been narrated to us by Ahmad Bin Muhammad, from Al-Hassan Bin Mahboub, from Saleh Bin Sahl Al-Hamdany and someone else, from Yunus Bin Zibyan, who has said: Abu Abdullah (as) said:

"When Allah (swt) Intends to Capture the soul of the Imam (as), and Creates an Imam (as) after Him, He Sends down a drop of water from beneath the Throne, to the Earth. It reaches the Imam (as) from a fruit or a vegetable when He eats that particular fruit or that particular vegetable, from whom Allah (swt) creates the seed of the Imam (as) who will stand after him (as)."

"Allah (swt) Creates from that drop (of water), the seed in the back, then Transfers it to the wife of the Imam (as). It stays there for forty nights. When forty nights pass by for Him, He hears the voice as if forty months have gone by. There is found written on His right shoulder, "And the word of your Lord has been accomplished truly and justly; there is none who can change His words, and He is the Hearing, the Knowing". (6:115)

"When He comes out on the Earth, He is Given the wisdom, and adorned with knowledge, and the dignity, and covered in prestige, and a lamp of Light is Made for Him by which He recognizes the conscience and He sees by it the deeds of the servants."

[Source: Basaair Al-Darajaat, Vol. 9, Chapter. 7, Hadees. 4]

وبهذا الاسناد، عن صالح بن عقبة، عن يزيد بن عبدالملك، عن أبي جعفر عليه السلام قال: لما ولدت فاطمة عليها السلام أوحى الله إلى ملك فأنطلق به لسان محمد صلى الله عليه وآله فسماها فاطمة، ثم قال: إني فطمتك بالعلم وفطمتك من الطمث، ثم قال أبوجعفر عليه السلام: والله لقد فطمها الله بالعلم وعن الطمث في الميثاق

Through the same chain of narrators it is narrated from Salih ibn ‘Aqaba from Yazid ibn ‘Abd al-Malik from abu Ja‘far (as) who has said the following.

"When Fatima (as) was born Allah sent an angel to make the tongue of Muhammad (saww) call her ‘Fatima’. Then he said, "I set you free (from ignorance) with knowledge and set you free from menses."

Abu Ja‘far (as) then said, "By Allah, He had set her free with knowledge (from ignorance) and from menses at the time of universal covenant (al-Mithaq)."

[Source: Al-Kafi, Kitab Al-Hujja, Chapter: 114, Hadees. 6]

الحسين [عن محمد] بن عبدالله، بن محمد بن سنان، عن المفضل، عن جابر ابن يزيد قال: قال لي أبوجعفر عليه السلام: يا جابر إن الله أول ما خلق خلق محمد صلى الله عليه وآله وعترته الهداة المهتدين، فكانوا أشباح نور بين يدي الله، قلت: وما الاشباح؟ قال: ظل النور أبدان نورانية بلا أرواح وكان مؤيدا بروح واحدة وهي روح القدس، فبه كان يعبد الله، وعترته ولذلك خلقهم حلماء، علماء، بررة، أصفياء، يعبدون الله بالصلاة والصوم والسجود والتسبيح والتهليل، ويصلون الصلوات ويحجون ويصومون.

Al-Husayn has narrated from (Muhammad) ibn 'Abdallah from Muhammad ibn Sinan from al-Mufaddal from Jabir ibn Yazid who has said the following.

"Abu Ja'far (a.s.) said to me, 'O Jabir, the first thing that Allah created was Muhammad (saww) and his offspring who are the rightly guided guides. They existed in 'Ashabah (form ) of light in the presence of Allah."

I then asked,"What is 'Ashabah?"

The Imam (a.s.) said, "It is the shadow of light, the lighting bodies without the spirit. It was supported by one spirit which is the Holy spirit. In that state he and his offspring worshipped Allah and for this reason He created them forbearing scholars, conscientious pure one who worship Allah through prayer, fasting, prostration, speaking of His Glory and the acknowledgment that He is the only Lord. They perform prayers, perform Hajj and fast."

[Source: Al-Kafi, Vol.1, Pg. 442, H. 10]

Note: We again see in the above narration which talks about the creation of Prophet Muhammad (saww) and the AhlulBait (as) and its very clear that Allah (swt) created them as Scholars (Ulama)

The chain of narrators should not be a problem here again as the content of this narration is repeated in various other narrations also.

علي بن محمد، عن سهل بن زياد، عن محمد بن علي بن إبراهيم، عن علي بن حماد، عن المفضل قال: قلت لابي عبدالله عليه السلام: كيف كنتم حيث كنتم في الاظلة؟ فقال يا مفضل كنا عند ربنا ليس عنده أحد غيرنا، في ظلة خضراء، نسبحه و نقدسه ونهلله ونمجده وما من ملك مقرب ولا ذيروح غيرنا حتى بدا له في خلق الاشياء، فخلق ما شاء كيف شاء من الملائكة وغيرهم، ثم أنهى علم ذلك إلينا

Ali bin Muhammad from Sahl ibn Ziyad from Muhammad ibn Ali ibn Ibrahim from Ali ibn Hammad from al- Mufaddal who has said the following.

"I asked Abu Abdallah (as), "How did you exist when you existed in the shadows?"

The Imam (a.s.) said, "O Mufaddal, we were in the presence of our Lord and there was no one else in the green shadow. We would speak of His Glory, Holiness, acknowledge Him as the Lords and speak of His Majesty. There was no angel even of the ones close to Allah or a living being except us in His presence until He decided to create all things. He then created whatever He wanted, in the He wanted of the angels and others then completed the knowledge of that to Us."

[Source: Al-Kafi, Vol. 1, Pg. 441, Hadith. 7]

This narration again shows that the AhlulBait (as) were given knowledge prior to coming to the world.

علي بن محمد، عن سهل بن زياد، عن موسى بن القاسم بن معاوية، ومحمد بن يحيى، عن العمركي بن علي جميعا، عن علي بن جعفر، عن أبي الحسن موسى عليه السلام قال: قال أبوعبدالله عليه السلام: إن الله عزوجل خلقنا فأحسن خلقنا، وصورنا فأحسن صورنا، وجعلنا خزانه في سمائه وأرضه، ولنا نطقت الشجرة وبعبادتنا عبد الله عزوجل، ولولانا ما عبدالله

Ali ibn Muhammad has narrated from Sahl ibn Ziyad from Musa ibn al-Qasim ibn Mu‘awiya and Muhammad ibn Yahya from al-‘Amrakiy ibn Ali altogether from Ali ibn Ja'far from Abu al-Hasan Musa (as) who has said the following.

Imam Abu Abdallah Sadiq (as) has said, ‘Allah (عز و جل) created us and He made our creation the best. He formed us and made our the best. He made us His treasurers in His heavens and His earth. For us the tree spoke and through Our worship، Allah (عز و جل), is worshiped. Had We not been there, Allah would not have been worshiped."

[Source: Al-Kafi, Vol. 1 Pg. 193]

[Note: Authentic Hadees as per Chain of narrations. Mirat Al-Uqool - Allama Majlisi Vol. 2, Pg. 349]

حدثنا احمد بن الحسين بن سعيد عن على بن اسباط عن ابيه اسباط عن سورة بن آليب قال قال لى أبو جعفر عليه السلام والله انا لخزان الله في سمائه وارضه لا على ذهب ولا على فضة الا على علمه.

Narrated to us Ahmad Bin Al-Husayn Bin Saeed, from Ali Bin Sabaat, from his father Asbaat, from Sowrat Bin Kaleyb who said:

Abu Jafar (as) said to me: ‘By Allah (swt), I am the Treasurer for Allah in His heaven and in His Earth, not for gold and not for silver, but of His Knowledge".

[Source: Basaair Al-Darajaat, Vol. 2, Chapter. 19, Hadees. 14]

حدثنا احمد بن محمد عن على بن الحكم عن داود العجلى عن زرارة عن حمران عن ابى جعفر عليه السلام قال ان الله تبارك وتعالى اخذ الميثاق على اولى العزم انى ربكم ومحمد رسولي وعلى امير المؤمنين عليه السلام واوصيائه من بعده ولاة امرى وخزان علمي وان المهدى انتصر به لدينى

Narrated to us Ahmad Bin Muhammad, from Ali Bin Al-Hakam, from Dawood Al-Ajaly, from Zarara, from Hamraan, who has said, Abu Ja’far (a.s) having said:

"Surely Allah (تبارك و تعالى) took the covenant from the Determined Ones (Ulul Azm Prophets); “I am your Lord, and Muhammad (saww) is My Messenger, and Ali (as) is Amir-ul-Momineen and his successors after him are the Masters of My Command (Wali Ul Amr), and Treasurers of My Knowledge, and through the Mahdi (atfs), My Religion will triumph".

[Source: Basaair Al-Darajaat, Vol. 2, Chapter. 19, Hadees. 14]

Imam al-Mahdi (as) reciting Quran on his Birth

ثم قال له: تكلم يا بني فقال عليه السلام: أشهد أن لا إله إلا الله وثنى بالصلاة على محمد وأمير المؤمنين والائمة حتى وقف على أبيه عليه السلام ثم قرأ " بسم الله الرحمن الرحيم ونريد أن نمن على الذين استضعفوا في الارض ونجعلهم أئمة ونجعلهم الوارثين ونمكن لهم في الارض ونري فرعون وهامان وجنودهما منهم ما كانوا يحذرون " ثم قال له اقرأ يا بني مما أنزل الله على أنبيائه ورسله فابتدأ بصحف آدم فقرأها بالسريانية، و كتاب إدريس، وكتاب نوح، وكتاب هود، وكتاب صالح، وصحف إبراهيم، وتوراة موسى، وزبور داود، وإنجيل عيسى، وفرقان جدي رسول الله صلى الله عليه وآله ثم قص قصص الانبياء والمرسلين إلى عهده

Imam Hasan Askar (as) said to Imam Az Zaman (atfs), "O My Son Speak". Imam Az Zaman (atfs) said, "There is no God but Allah, and praise and blessings be upon Muhammad (saww) and Amir Al-Momineen (as) and the Imam's (as) .

Then Imam (atfs) recited the verse:

"And We wished to be Gracious to those who were being depressed in the land, to make them leaders (in Faith) and make them heirs. To establish a firm place for them in the land, and to show Pharaoh, Haman, and their hosts, at their hands, the very things against which they were taking precautions." (28:5-6)

Then Imam Hasan Askari (as) said, "Read my son from what Allah (swt) send down upon the Prophets (a.s) and Messengers (a.s).

Then Imam Az Zaman (atfs) started with the recitation of the book of Adam (a.s) in Syriania Language and book of Idrees (a.s) and book of Noah (a.s) and book of Hud (a.s) and book of Saleh (a.s) and book of Ibraheem (a.s) and Taurat of Moosa (a.s) and Zaboor of Dawood (a.s) and Injeel of Isa (a.s) and Furqan (Quran) of Prophet Muhammad (saww) and then told the stories of the Prophets upto His time.

[Source: Bihar Al Anwar Vol. 51 Pg. 26-27]

Imam Ali (as) reciting Quran and more on his Birth

حدثنا الشيخ أبو جعفر محمد بن الحسن بن علي بن الحسن الطوسي رضي الله عنه، قال أخبرنا أبو الحسن محمد بن أحمد بن الحسن بن شاذان، قال حدثني أحمد بن محمد بن أيوب، قال حدثنا عمر بن الحسن القاضي، قال حدثنا عبد الله بن محمد، قال حدثني أبو حبيبة، قال حدثني سفيان بن عيينة، عن الزهري، عن عائشة. قال محمد بن أحمد بن شاذان و حدثني سهل بن أحمد، قال حدثنا أحمد بن عمر الربيعي، قال حدثنا زكريا بن يحيى، قال حدثنا أبو داود، قال حدثنا شعبة، عن قتادة، عن أنس بن مالك، عن العباس بن عبد المطلب. قال ابن شاذان و حدثني إبراهيم بن علي، بإسناده عن أبي عبد الله جعفر بن محمد عليهما السلام عن آبائه

...

فلما أردت أن أخرج و ولدي على يدي هتف بي هاتف و قال يا فاطمة، سميه عليا، فأنا العلي الأعلى، و إني خلقته من قدرتي، و عز جلالي، و قسط عدلي، و اشتققت اسمه من اسمي، و أدبته بأدبي، و فوضت إليه أمري، و وقفته على غامض علمي

...

ثم دخل رسول الله )صلى الله عليه و آله(، فلما دخل اهتز له أمير المؤمنين )عليه السلام( و ضحك في وجهه، و قال السلام عليك، يا رسول الله، و رحمة الله و بركاته. قال ثم تنحنح بإذن الله )تعالى(، و قال بِسْمِ اللَّهِ الرَّحْمنِ الرَّحِيمِ قَدْ أَفْلَحَ الْمُؤْمِنُونَ الَّذِينَ هُمْ فِي صَلاتِهِمْ خاشِعُونَ إلى آخر الآيات. فقال رسول الله )صلى الله عليه و آله( قد أفلحوا بك، و قرأ تمام الآيات إلى قوله أُولئِكَ هُمُ الْوارِثُونَ الَّذِينَ يَرِثُونَ الْفِرْدَوْسَ هُمْ فِيها خالِدُونَ

[Quoting only a part of a very lengthy narration]

Narrated from Imam Jafar Al-Sadiq (as) from his fathers:

...

Fatema bint Asad (sa) says,

"When I wanted to come out, a caller called me and said, “O Fatema! Name him Ali because he is Ali (High) and Allah is Ali Al-A’la (the highest). Allah tells you, ‘I derived his name from My name, And I educate him with My manners, and authorized my matters to Him and I taught him My most obscure knowledge."

....

Then the Holy Prophet (saww) entered and when He entered, Amir Al-Momineen moved and smiled in His face and said, "Peace be upon You O'Prophet of Allah (saww) and Allah's (swt) Mercy and Blessings be upon You.

Then He [Imam Ali (as)] made a sound of 'Hem' and said, " In the Name of Allah, the Beneficent the Merciful. The believers indeed are successful. Those who humble themselves in their prayers." (23:1) to the last verses.

Then Holy Prophet (saww) said, "Indeed success is with You". And then recited all the verses upto His Saying, "These will be the heirs, Who will inherit Paradise: they will dwell therein (for ever). (23.9-10)

[Source: Al-Amaali - Sheih Toosi, Pg. 706]

This narration clearly shows that Imam Ali (as) was born with the knowledge of the Quran and both Imam Ali (as) and Prophet Muhammad (saww) had knowledge of the Quran even before its revelation.