bartik_links__system_main_menu() doesn't work
Make sure that:


theme('links__system_main_menu', ... is actually used to output the menu.
There are no issues with your code.
Caches have been cleared after adding the function (admin/config/development/performance).


To better understand what system_main_menu is all about check keithm's answer here. The main part of the explanation is this:


  links__system_main_menu is a theme hook pattern of the form [base
  hook]__[context]. When links are themed with
  theme('links__system_main_menu', $vars), theme() will search for
  and use theme_links__system_main_menu() if it has been defined. If
  not, it will use theme_links().


Official documentation about it can be found here.