Ultimately how much proof you provide is up to you. If you're trying to avoid a lawsuit, the more the better. That is, it's not as though you're "tipping your hand" in showing this evidence to the employee.
Discharges in Montana are governed by the Montana Wrongful Discharge from Employment Act (“WDEA”). Under that statute, employee terminations are limited to terminations for “good cause.” The WDEA defines “good cause” as “reasonable job-related grounds for dismissal based on a failure to satisfactorily perform job duties, disruption of the employer’s business operation, or other legitimate business reason.” MCA, Section 39-2-903(5). In turn, Montana courts have explained that a “legitimate business reason” means a “reason that is neither false, whimsical, arbitrary or capricious, and it must have some logical relationship to the whims of the business.” In short, as the WDEA and the decisions interpreting the statute make clear, Montana employers do not have the flexibility afforded employers in most, though not all other states, to terminate an employee at any time for any legal reason.
So in short, showing that the employee engaged in lying and other for cause termination reasons would be enough. Technically you don't have to show the employee anything, but that might make it more likely that the employee will sue. Showing that you actually have this evidence would make the employee less likely to sue. So while it's ultimately up to you how much you show, I would suggest that you show more, rather than less, to show that you are not just bluffing about the evidence that you do have.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Thank you, ***** ***** luck to you!