Funny how often in a lawsuit one party makes an allegation which is challenged by the other party, then comes a dispute as to which party is to prove its claim ... which party has the burden of proof ... Just like a pingpong ball bouncing back and forth...
Let us see how native professionals (UK or US judges, attorneys etc.) speak about burden of proof:
(1) person + has + burden + to prove + fact
For example: A person charged with an offense has no burden to prove innocence.
(2) prosecution + has + burden of proof + to demonstrate + that + subject + verb
For example: In a criminal trial, the prosecution has the burden of proof to demonstrate that the defendant committed the specified crime(s) beyond a reasonable doubt.
(3) burden of proof + is + on person + to verb1
For example: The burden of proof is on the plaintiff to establish that s/he was the victim of unlawful discrimination.
(4) burden of proof + rests + on person + to verb1
For example: The burden of proof rests on the plaintiff to establish that he or she will be an adequate class representative.
(5) burden of proof + lies + with person
For example: The burden of proof lies with the party requesting the hearing.
(6) burden of proof + shifts + from person + to person
For example: The burden of proof will shift from the complainant to the respondent if the complainant can prove facts.
(7) burden + of verb+ing + is borne + by person
For example: The burden of proving intentional discrimination is borne by the plaintiff.
The term "burden of proof" is described in the book entitled 'A Practical Guide to English for Law'. It is also described in PDF ebook entitled 'Terminology of Civil Procedure in Use'.