What is the evidence for the permissibility of discontinuing a treatment that is expected to prolong life? And is that considered casting oneself into destruction or neglecting those under one's care?

Medical treatment is not obligatory unless its benefit is certain, or if abstaining from treatment leads to perishing, such as stopping a hemorrhage or amputating a damaged limb whose retention would lead to the decay of the rest of the body. The Islamic Fiqh Academy has adopted the view that treatment is obligatory "if abstaining from it leads to the destruction of the soul or one of the organs, or to disability, or if the disease's harm can be transmitted to others." The described case does not fall under obligatory treatment, as there is no certainty of perishing. Rather, doctors say that the person will live but their lifespan will be shortened, and this is not certain destruction. Therefore, treatment in this case is permissible, and there is no blame on the father if he chooses to forgo it and suffices with what alleviates pain and suffering, especially if the treatments are harsh.