When trying to prove a proposition we usually either try to prove by the result of some related theorems, or try some "tricks" that often work for "similar" types of problems. Textbooks in mathematics are often structured by definitions and theorems (and lemmas, etc.), and sometimes some explanation of the intuition behind theorems, but usually authors won't explicitly mention certain techniques/tricks that they used in the proof, and it takes time to understand what they did in certain parts of the proof. I'm wondering why didn't anyone (at least to my knowledge) make a systematic listing of such strategies/techniques/tricks, as they are also important to mathematics.
Also has there been any examples where something that used to be an informal trick later becomes a formalized theorem once we know exactly what is the condition to apply the trick? Do all tricks have the potential of becoming theorems?
Posting in real analysis because I don't really know where I should post, and this is the field where such tricks seem most commonly used.