If the INS was not updated at the gate first flight of day
Must go back to the original question. For the first flight of the day you must turn the INS on. It would never be left on on an unattended aircraft. If the ground power failed then the INS would so go pear shaped with no cooling, and would be running on the aircraft (or its own) battery.
When you turn on an INS you must enter the correct coordinates or the ALIGN light will start flashing at you.
But I bet most short haul pilots never fly on the INS. There is even a Caution message on the A320..INS NAV.. to tell you that you have lost DME updates.
Also on an A320, there is no INS. There are three IRS sensors that supply position to the FMS, along with the 2 GPS and the 2 DME and 2 VOR. The FMS decides which sensors to use and starts with DME/DME.