Good luck trying to prove your position, as it’s hard to show there was no oil added. Their position will be that the compressor is 5 years old and died a natural death. Check your paperwork from the warranty repair for mention of oil being added either in the narrarive of what was done, or in the lists of parts and supplies used.
If no oil is listed, point this out to the repair facility, and they may accept liability. If oil is listed, your position is harder to prove. Even if your paper work states oil was added, it may not have been.