1. An information processing unit having an auto power-off function, comprising: a mode setting means for setting one mode from a plurality of pre-prepared power-saving modes in conformity with input by a user, wherein the plurality of power-saving modes includes a first mode and a second mode, and the first mode and the second mode differ in electricity consumption reduction effectiveness; a determination means for determining whether a non-operated state in which no input operation has been carried out by the user has continued for a predetermined time or longer; a processing execution means for executing internal processing; and a decision means for deciding whether or not to execute the auto power-off function based on the mode set by the mode setting means and internal processing executed by the processing execution means when the determination means has determined that the non-operated state has continued for a predetermined time or longer, wherein the decision means decides not to execute the auto power-off function when the mode setting means sets the first mode and the determination means has determined that the non-operated state has continued for a predetermined time or longer while the processing execution means is executing predetermined internal processing, the decision means decides to execute the auto power-off function when the mode setting means sets the second mode and the determination means has determined that the non-operated state has continued for the predetermined time or longer while the processing execution means is executing the predetermined internal processing, the predetermined internal processing is game application execution and video replay; wherein the internal processing executed by the processing execution means is classified into a first group, a second group, and a third group; when the determination means has determined that the non-operated state has continued for the predetermined time or longer while the processing execution means is executing internal processing belonging to the first group, the decision means decides to execute the auto power-off function even in the case where the mode setting means sets either one of the first mode or the second mode; when the determination means has determined that the non-operated state has continued for the predetermined time or longer while the processing execution means is executing internal processing belonging to the second group, the decision means decides not to execute the auto power-off function in the case where the mode setting means sets the first mode whereas the decision means decides to execute the auto power-off function in the case where the mode setting means sets the second mode; and when the determination means has determined that the non-operated state has continued for the predetermined time or longer while the processing execution means is executing internal processing belonging to the third group, the decision means decides not to execute the auto power-off function even in the case where the mode setting means sets either one of the first mode or the second mode.