An “infallible” teaching of the Pope, of which there has been exactly one since Vatican I, is Christ speaking directly through the Pope as heir to Peter's grant of the Earthly keys to the Kingdom.
IOW, an infallible teaching isn't really the Pope's. It is His. And as stated, it is beyond very rare.
Seen in that light, much of the to do about the Pope being “infallible” is wasted breath.
Do tell.
Actually Ordinatio Sacerdotalis on the question of women and priesthood was clearly irreformable. The libs asked the Congregation for the Doctrine of the Faith in a “dubium,” whether John Paul had intended that it be understood as infallible and the dubium was answered with “Yes.” Then the libs complained that the CDF lacked competence to rule on infallibility, only the pope does—so why did you doofuses send in a dubium? Only when you didn’t get the answer you hoped for do you decide the CDF lacked competence.
In Evangelium Vitae (1995) John Paul II made three clearly, beyond any doubt, infallible statements:
1. that taking of innocent human life is always wrong and no good intention can make it good. Look at par. 52 or 57 or somewhere in there. He clearly invokes the authority of Peter, acts in communion with all the bishops, points out that Scripture, unbroken teaching, and natural law all confirm this truth, then explicitly claims that he is making a definitive ruling. He had to claim the highest level of authority here because some people, for the first time in history, were arguing that sometimes taking of innocent life is okay. Always before, people justified taking of life by first declaring the one(s) being killed guilty. War and capital punishment are not exceptions to no. 1 because they take guilty life. They should be avoided as much as possible but if ever justified, they are cases of taking of guilty life. So there are no exceptions to taking innocent human life.
2. that abortion is a case of taking of innocent human life. Again, some people argue that it’s not human or not innocent and he’s saying, yes it is. Period.
3. that euthanasia also is a case of taking innocent human life (euthanasia defined as deliberately, not accidentally or double-effect, taking the life of a born person. Again, some argue that compassion or some other good intent (mercy killing) makes taking of innocent human life okay. He said, “No.”
and each of the three was as clear an example of an irreformable teaching as you will ever want to see. I use these three to try to clarify for students just what an infallible statement looks like.