Mince Pies

A mathematical Grandmother left a plate of mince pies on the table with her four grandchildren, and came back to find it empty!

Art said “Bert and I ate the same number of pies between us as Caz did by herself.”

Bert said “Art ate the same number as Diz and I together had”

Caz said ‘The number of pies I ate plus twice Diz’s number is equal to twice Art’s number plus Bert’s

Diz just smiled, because she knew Granny had left the smallest number of pies for these statements to all be true.

How many pies did Granny leave on the plate?