I would advise against consuming a tub of ice cream that has been melted, partially eaten, and then put back in the freezer for over a week.
When ice cream melts and is then refrozen, it can undergo changes in texture, consistency, and taste. The formation of ice crystals during the initial freezing process and subsequent melting and refreezing can negatively impact the quality of the ice cream. It may result in a grainy or icy texture and a loss of the creamy, smooth consistency typically associated with ice cream.
Additionally, storing partially consumed food items for extended periods increases the risk of bacterial growth and foodborne illnesses. Once ice cream has melted, it becomes a potentially favorable environment for the growth of bacteria. Refreezing it may not eliminate the bacteria that may have multiplied during the period it was in a melted state.
To ensure food safety and the best quality, it's generally recommended to discard any perishable food item that has been left out at room temperature or has gone through multiple cycles of melting and refreezing. It's better to prioritize your health and consume fresh ice cream or discard the melted and refrozen portion.