Proof is a 2005 American drama film directed by John Madden and starring Gwyneth Paltrow, Anthony Hopkins, Jake Gyllenhaal, and Hope Davis. The screenplay was written by Rebecca Miller and David Auburn and based on Auburn's play, Proof.

Robert, a brilliant mathematician who used to work at the University of Chicago, startles his daughter Catherine while she watches TV. He gives her a bottle of champagne for her birthday, and they chat. All of this turns out to be a dream; Robert died the previous week, after a long period of crippling mental illness, and his funeral is tomorrow.

Meanwhile, Hal, a former graduate student of Robert's, is reading through the latter's notebooks, which are filled with meaningless notes. Hal believes that Robert's genius may have withstood his mental illness, and clues to that genius might lie in one of his notebooks. When Hal comments on the vast amount of work Robert did, a suspicious Catherine searches Hal's backpack. A notebook eventually falls out of his coat. He explains that he wanted to give the notebook as a birthday present because it "had something written in it about her". Hal is forced to leave, giving the notebook as intended, when Catherine calls the police.

The next day, Catherine's sister, New Yorker Claire, arrives in town. At the funeral, Catherine berates the many attendants for not being there for Robert during his descent into insanity. She ends by saying that she is glad her father died and leaves mid-funeral.

Claire decides to sell Robert's house back to the university and wants Catherine to come with her to New York. A wake held at the house the night is attended by academic mathematicians. Hal appears and chats up Catherine. Softening up to Hal, Catherine has emotional sex with him.

In flashbacks, Robert is shown invigorated, believing that he has seen the beginnings of a new mathematical proof that will prove his triumph over mental illness. In the present, Catherine gives Hal a key to Robert's desk and tells him to check the locked drawer for a notebook which contains an important proof. Excited, he shows the discovery to Catherine and Claire. He asks how long Catherine knew about this and why she did not mention it. Catherine, a promising mathematician herself, says that she wrote it and not Robert, despite evidence to the contrary. Neither Hal nor Claire believe Catherine. Hal believes that the proof's mathematics are beyond Catherine, while Claire suspects that Catherine is suffering the onset of mental illness. Hal decides to take the notebook to the math department to verify the proof's accuracy.

He eventually returns as Claire and Catherine are leaving, with news that the math department believes the proof to be valid. Hal does not think that Robert wrote the proof because it employs newer mathematics and wants Catherine to explain it. Catherine remains stung by his earlier lack of trust, and the sisters leave for the airport. Hal sprints after the car and throws the book through the window and onto Catherine's lap.

In another flashback, it is revealed that, while living together, Robert challenged Catherine to work on math, which she does, ultimately completing a proof, which she describes in one of the house's notebooks. Catherine goes to tell Robert about the breakthrough, but he insists that she read aloud the proof that he is working on. To Catherine's disappointment, Robert's notebook contains only ramblings. Reading his work, Catherine realizes that Robert has not overcome his mental illness.