...[Y]ou are on a mountain, and I attack you and throw you into a place filled with rocks that will tumble down below if given a push. The only way you can leave that spot is by causing a landslide. On a perch below is someone else I also forced there. If there's a landslide, their perch will be destroyed and they'll fall to their death. There is no way for your calls for help to be heard; you have to wait until you are discovered missing and a rescue party is sent. Let's assume that you are in no danger; one of your hobbies is to be a survivalist; you know how to attract game birds. You are able to live off them until found, but that will take nine months. The other person is also able to survive because your efforts to attract birds will inevitably attract birds to their perch, too.
Does your right to liberty include a right to push the rocks out of the way and cause the death of the other person?
It is also important to point out each right is predicated upon the previous one(s). You can't pursue happiness if you aren't free and you can't be free if you're not alive and you cannot pursue happiness if you'ren't alive.