What evidence do Francis, Giles, and John have that the girls are lying?
For the most part in Act III, the three men offer analysis that undermines or at least forces one to examine the basis of spectral evidence that the girls have offered. Francis Nurse's petition is a document that can serve as evidence against the girls. It is a statement signed by 91 different people that neither Rebecca Nurse or Elizabeth Proctor are witches or engage in witchcraft. Corey is a part of this as well, suggesting very clearly and openly that the trial is not about witchcraft. Rather, it is about people like Putnam being able to garner more land and engineer their ability to take land from those who have been accused. Proctor arrives with Mary Warren, one of the girls in question, to offer testimony that Abigail and the other girls are operating out of the desire for power and control and do not represent the truth. In response to each of the mens' claims, the court offers the idea that anyone who speaks out against the court, and thus the girls' accusations, seeks to bring down the court. This "either you are with us or against us" mentality is how Danforth that court align themselves against the evidence and analysis that the three men in Act III offer as counter to the girls' claims of witchcraft.
First, John Proctor has his own testimony; Abigail Williams told him that the girls' illness has nothing to do with witchcraft, that Reverend Parris simply surprised the girls in the forest and Betty got scared. Proctor knows that he may need to share this information with the court in order to exonerate his wife and his friends' wives. John also knows that he may have to confess to his affair with Abigail in order to prove that she has a motive to accuse his wife of witchcraft. This testimony would serve as evidence against her as well.
Second, Proctor, Corey, and Nurse have the testimony of Mary Warren, a girl who admits that all the girls have been pretending only.
Thirdly, Corey has his deposition which states that an honest (but anonymous) man overheard Mr. Putnam admit, essentially, that he put his daughter up to accusing George Jacobs of witchcraft so that Putnam could purchase Jacobs' land when it goes up for auction. However, Corey refuses to tell the magistrates the name of the man who has given him this information, and so his deposition is disregarded and he is held in contempt of court.