Up to this point I have been dealing with the creation of Adam as an allegorical story of moral values, as mentioned in the Chapters Al-Baqarah and Al-A’raaf, which portray the earthly material state of humanity, a humanity created in the most perfect fashion by Almighty God (swt), and left needing nothing in all aspects of life. God sheltered Man in the life of moderation and advised him against following his worldly desires, transgressing into a life of extravagance and wastefulness. Man is also advised against forgetting the godly, spiritual aspect of life and the covenant, and not to heed to the whisperings of Satan, who decorates for him a world without a Lord, filled with the sweetness of endless pleasures, promises positions of power and hegemony over all creation. Until such point when he comes under the authority of Iblis and the control of these temporal pleasures, and all he can do is try to cover his shame with more degradation and dishonour, ending with his expulsion from the worldly paradise to the abode of hardship and disappointment.
But to establish and prove the general infallibility of prophets, we need to look at this story as an actual occurrence in history. Since Adam’s progeny required a template of human perfection as a goal, he was not only God’s first human creation, but also a Prophet to his offspring. This means that infallibility is a necessary part of his existence.
To understand the situation, we need to define ‘prohibition’ (نهي). What kind of prohibition is intended here? Is it meant to be a forbiddance to do evil acts (نهي مولوي)? Or is it comparable to a doctor advising you not to eat certain types of food in order to prevent a life-time of ailments and a deterioration in your health, in the future (نهي إرشادي)?
Indeed, it was the latter, considering the following facts:
Adam and Hawa (Eve), were placed in a temporal abode of bliss (not the heavenly paradise, since creations like Satan, and matters of luring and evil whisperings have no authority in such a place), and since their destiny was to be God’s representatives on Earth, they had to satisfy certain qualities (explained in previous posts), in order for the Angels to accept him as their superior. Therefore, on account of his superiority to Angels, it makes no sense that such a being can succumb to evil whisperings and worldly lusts.
Adam repented, but was not sent ‘back’ to heaven, which again proves firstly that he was not in heaven to begin with, and that his destiny remained to be God’s viceregent on Earth.
The jurisdiction of the religious law upon them did not start until they fell to the earthly world, since the prohibition would result in hardship on Earth in the form of having to provide for his family food and shelter etc, and not be a sin in itself which would result in a distancing from God in the spiritual sense.
In the Qur’an there exist decisive verses (محكمات), where there can be no misinterpretation or re-interpretation, and allegorical verses (متشابهات). It is the decisive verses that confirm and dictate the allegorical ones. The allegorical ones are derived from the decisive verses. Therefore, since it is a cardinal rule in the school of thought of the Ahlul Bayt (progeny of the Holy Prophet), that all Prophets (peace be upon them) are infallible from any kind of sin, then it would be inconceivable that Adam (as) would disobey an obligatory command. ‘And verily We made a covenant of old with Adam, but he forgot, and We found no constancy in him  (وَلَمْ نَجِدْ لَهُ عَزْمًا)’ . His guilt was that he did not demonstrate constancy (steadfastness, will-power), not that he violated Allah’s commands, because the commands were advisory and not obligatory.
In Christianity, the fallibility of Adam is considered to be the cardinal doctrine of this faith. If Jesus was not sacrificed in the atonement for the sin committed by Adam (which God required in order to forgive our sins), there would be no basis for Christianity.
The Islamic concept of infallibility is an extensive one and requires separate discourses, therefore I will dedicate a few posts to this topic when I tackle the concept of Prophethood and Imamate according to the school of thought of the Ahlul Bayt.
 See Holy Qur’an: [Al Imran:33], [Al-Hijr:40-42], [Maryam:51]
 ‘He it is Who hath revealed unto thee (Muhammad) the Scripture wherein are clear revelations – they are the substance of the Book – and others (which are) allegorical. But those in whose hearts is doubt pursue, forsooth, that which is allegorical seeking (to cause) dissension by seeking to explain it. None knoweth its explanation save Allah. And those who are of sound instruction (the Prophet and Ahlul Bayt).’ See commentary of this verse in the Holy Qur’an [3:7], here.
 Infallibility means protection. In Islamic terminology it means the spiritual grace of Allah (swt) enabling a person to abstain from sins by his/her own free will. The power of infallibility or without sin does not make a person incapable of committing sins, rather he/she refrains from sins and mistakes by his/her own power and will.
 Holy Qur’an [20:115]
 Deeper analysis of this concept can be found here.