Although I have observed that "the" is not generally used before the word "proof," it is still unclear why it is not used even if it is an "of phrase." For example: "we have to procure proof of his involvement in the crime." The word "proof" is modified there by "of his involvement," so why is the definite article is not used there like: "We have to procure the proof of his involvement in the crime"?
You can use the with proof, but only if you know there's only one proof (one fact or circumstance that provides proof) or if you want to restrict the discussion to one proof.
The proof you stole the cookies is the crumbs on your chin.
If you're uncertain how many different ways there might be to prove something, then you would omit the as in your examples.
Proof is being used uncountably, because it may or may not be one "thing" that proves something - we simply don't know so it's "fluid."
Uncountable singular nouns don't take articles.
We have to procure water for the plants.
We have to procure proof of his involvement.
It's possible to use proof in a countable manner, and the meaning is different - typically "an argument that proves something", or "test copy of a document or other printed work."