Can the Pope Become a Heretic?

Let us divide this down into the various sub questions.

  1. Can the Pope Become a Heretic as a Private Doctor?
  2. Can the Pope Become a Heretic in his public actions as Pope?
  3. Has any Pope Become a Heretic?
  4. What Would Happen If the Pope Became a Heretic?

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If A Pope Becomes A Heretic Does He Automatically Fall From Office?