To give an example, there's an infinite array of "If X is greater than 20, X is greater than 10" theorems. Obviously useless to prove each of them, but enumeration will encounter all of these, as well as a huge number of classes of similar theorems, in the process of getting to anything interesting.
You can't just "skip" them either, because they get arbitrarily complicated. I chose a really simple one to drive home the point, and because it's the sort of theorem you'll hit early, but tautologies can be arbitrarily complicated. Trying to create something that identifies them hits halting problems of their own. Rice's Theorem, which can be colloquially summarized as "any interesting property is not computable", will stop you, among other things.
("If X is a prime > 521, X does not have 21 as a factor." "If X is irrational, its fraction expansion does not have a denominator between 15 and 54." "If the absolute value of X is not equal to X, X < 932." "If the fractional component of X is greater than .5, the first digit of the fraction expansion is not 2 or 4." "The sum of two positive numbers is greater than the smallest of the two numbers number minus 273,883,192,823." "Most" theorems are really, really useless.)
I think you meant in the opposite order, i.e.: "If X is greater than 20, then X is greater than 10" :)
But otherwise your point is valid.
I think I have seen theorems of that sort. (Can't think of a specific example right now.)
You could only prove it if you add something else, e.g. a precondition saying that X = 30.
I can see how it would fall apart quickly for something like the Riemann Hypothesis, because you're searching not for an equation (well, if you found a counterexample, sure) but for a nebulous classification problem: all solutions to this equation ( this series of tokens from this language that return zero) must lie on (I can't recall if it was literally on, or "around") re(1/2). Once you try to reason with infinite sets (domain/codomain) the logic of a CPU can't help you nearly as much.
If you are going to do something more clever than "generate all tokens", then you're no longer talking about the same thing, and the analysis depends on the nature of your "more clever". Although starting with an exponential (or super exponential) process and then trying to "cut it down" to something reasonable is almost always a fool's errand anyhow. Starting yourself out in the position where you are not only behind the eight ball but have essentially already been murdered by it, and then cleverly working yourself back into a position where you've somehow won, is rarely, if ever, a good approach to solving a problem.
I'm just thinking aloud, but suppose a random-math-syntax shuffler eventually produced the quadratic formula, how would you even check it? You would have to input many permutations of quadratic equation inputs and check them against the randomly-produced one. You'd quickly hit imaginary numbers, which could (falsely) signal that the QF is invalid.
Compare that to monkey-typing fizzbuzz. That seemingly could be done in a few billion iterations, no? (assume you could exclude monkeys from typing syntax errors, and only produce valid C++/java whatever).
IDK, I'm just mesmerized by the fact that the QF is so compact, yet there isn't an easily-discoverable way to derive it by brute force. (and I'm deliberately excluding AI because of how tiring the questions of the form "why can't AI do _____" are. I don't care that much about AI).
A proof of the theorem is going to be at least a few hundred bits, which is totally out of reach of brute force search.
A 10 billion search space only covers 34 bits (less than 5 ASCII symbols) in which you'll struggle to even state the theorem.
Even 12 tokens with only 10 different token values gives you 10¹² different candidate ‘proofs’. That already is a thousand times 10 billion.
In reality, there easily are 100 possible token values (lower and upper case alphabet, punctuation, Greek lower and upper case, dozens of symbols, a bit of Hebrew, etc), taking that number to 10²⁴.
Also, I think even just express the two solutions to ax²+bx+c=0 without any hint of a proof in twelve tokens is an insurmountable challenge.