When to say "a proof", "the proof" and just "proof"?

Solution 1:

Your friend's sentence is both grammatically and idiomatically correct.

In the sentence you quote, "is proof" is closer in meaning to the verb "proves" than it is to "is {a / the} proof".

In this sense it is a fairly rare type of usage, but "is proof" with the meaning "proves" is a normal expression.

Solution 2:

Normally, 'a / one proof' is used where it is acknowledged that there are different sufficient proofs (ie a count noun usage):

One proof of Pythagoras's Theorem uses constructions and congruent triangles. A proof that is easier for many to understand uses tessellations.

'The' can be used with both count (when it makes sense to single out that proof) and non-count usages:

The proof using congruent triangles was the one most usually taught fifty years ago in schools.

The proof of Fermat's Last Theorem is considerably more difficult.

Article-less 'proof' is again a non-count usage, used in more general contexts:

Proof is necessary for a theory to be fully accepted.

His fingerprints in the room are proof that he is lying.

Solution 3:

"A/the proof" is most commonly used to refer to an actual formal mathematical construction, i.e. a proof of a mathematical theorem.

As Erik noted, your friend's sentence is correct, but it is the more informal use of the word 'proof,' meaning 'evidence.' When used in this sense, the article is usually excluded.

Really, the word 'evidence' would have been a better choice here, but 'evidence' and 'proof' have unfortunately become conflated in modern usage. I say it is unfortunate because the formal usage actually refers to a related but quite different concept. With a proof (in the formal sense of the word,) if the proof is constructed correctly, it is not possible for the conclusion to be false if the premises on which it is based are true.

Proofs require either deductive reasoning or exhaustive induction to be used. Evidence, on the other hand, can be interpreted to suggest a conclusion, but doesn't actually prove that conclusion to be necessarily true. Such a conclusion is reached via inductive reasoning. This seems to be the case with your friend's assertion.