Was following the father, when there was evidence of his breaking the fast, a correct deliberate act, or is it considered a mistake made with the intention of changing the reprehensible? Is it sufficient to state the ruling without direct advice, or is it obligatory to inform him of what he saw to absolve himself? And is the son's admission of error considered a lie from him after his father accused him of lying, despite his certainty of his father's action? What is his duty in this situation?

You erred by spying on your father, and you must think well of him. Your confession to him about your mistake is not a lie. Clarifying the Shar'i ruling is sufficient, but your bad conduct towards him is forbidden, and the right of a parent is great. Enjoining good and forbidding evil for him should be done with politeness and gentleness, not with violence, abuse, or harsh words; otherwise, you should leave him be, for a father is not like a stranger. You must honor your father and be kind to him, and one of the greatest forms of honoring is to help him repent and become righteous.