Here's the general rule when it comes to articles: Singular indefinite - A/an + noun Singular definite - The + noun Plural indefinite/indefinite and uncountable - (no article) noun Plural definite/plural and uncountable - The + noun I'm guessing that French is like Italian in that all plural nouns use an article, regardless of whether or not they're definite. In English, the distinction between definite and indefinite nouns is as important for plural or uncountable nouns as it is for singular nouns. "Proof" is indefinite and uncountable here, so there's no article. If the speaker had already been looking for proof, and the listener knew, then you'd say "the proof" instead.

Yet and not just with the word proof. You don't need an article whenever a noun follows the verb "to be." So you say, "I am happy" not, "I am the happy." You say, "he is tall" not, "he is the tall." In exactly the same way you say, "this agreement is proof" because proof is a characteristic of the agreement.

Yes, 'proof' is often an abstract noun, and therefore uncountable. To confirm that something is true, you need to have 'proof' or 'some proof'.