Are drink cups from fast food compostable?

Hi there,

I have usually added drink cups from say  once shredded into compost. I have been finding a plastic layer which seems like the only source could be from these cups. Do they add a hidden layer of plastic between the paper layers or something?

I have noticed food containers like ice cream boxes have this plastic layer and I no longer add those types of containers. It is weird as once I am tearing the cups into small pieces, I never feel the resistance of plastic.Why do not they have some from biodegradable cups wholesale?

Normally if I rip it apart slowly I'd see the pull of plastic but with the cups I never see it. I assumed the cups were paper with a wax layer. Help me solve this mystery.