First Stolen Notebook (Situational Irony)
Hal drops one of Robert's notebooks as he's leaving after insisting that he has stolen nothing. At first, Catherine believes he is stealing it to plagiarize the work, but it turns out that, ironically enough, he is taking it in order to wrap it and give it to her as a birthday gift because it contains a sweet note written by her father about her. The irony lies in the fact that Catherine is suspicious about something that is actually a sweet gesture.
Robert is a figment (Situational Irony)
In the very first scene of the play, it seems as though Catherine is talking to her father, Robert. Towards the end of the scene, in an ironic twist, it is revealed that Robert has been dead for a week, and Catherine is actually speaking to a figment of her imagination, a vision of him.
Hal taking the notebook with the proof (Dramatic Irony)
After Claire and Hal do not believe her about writing the proof, Catherine takes to her bed and becomes catatonic, entering a deep depression. While she is bed-ridden, Hal visits the house and Claire gives him the notebook with Catherine's proof in it, without Catherine's permission. This creates dramatic irony in that the audience knows something significant that Catherine does not.
Catherine wrote the proof (Dramatic Irony)
In a flashback, we see that in his declining years, Robert was plagued by mental illness, and was in no state to write a complicated proof. Because the audience sees these illuminating flashbacks, they become aware of something the other characters don't know: that Catherine was the only one who could have written the proof.
