Those caught without valid proof of payment are subject to a fine.

It is not known whether Fermat had actually found a valid proof.

For Ben, admitting that he has a fear of peacocks is valid proof.

Joe has been able to see valid proof of the success of his Cold Case reports.

You will need to support all preference claims with a valid proof of preferential origin.

After this operation the obtained graph is transformed into a valid proof.

Because of soundness, we know that the user must really act honestly in order to be able to provide a valid proof.

This vote is valid proof that we are stronger as a nation when we work together.

But valid scientific proof must be accepted even when it gives unacceptable answers.

As in natural deduction, theorems are those where is the conclusion of a valid proof.