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Proof (Noun)

The act of validating; finding or testing the truth of something.

Nouns denoting acts or actions.

Any factual evidence that helps to establish the truth of something; "if you have any proof for what you say, now is the time to produce it".

Nouns denoting cognitive processes and contents.

The police officer asked the witness to provide proof of her identity before taking her statement.
After months of searching, the scientist finally found the proof she needed to confirm her groundbreaking theory.
He couldn't find any proof that his ancestors had actually been knights in the 12th century.
The court required proof of the defendant's guilt before it could convict him of the crime.
If you have any proof of plagiarism, you need to report it to the academic integrity committee immediately.

A formal series of statements showing that if one thing is true something else necessarily follows from it.

Nouns denoting communicative processes and contents.

Euclid's Elements includes a geometric proof that the sum of the interior angles in a triangle is 180 degrees.
The proof relies on several lemmas, but the core argument is straightforward and elegant.
The mathematician spent years working out the proof, often going back to the beginning when she found a flaw.
The lecturer presented a constructive proof that the square root of two is irrational.
The theorem's proof involves induction on the height of the tree, with the base case being a single node.