Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$ i.e. Horn's inequalities for n matrices?

Can such inequalities be obtained just from Horn's inequalities for 3 matrices if we couple matrices in this way $$(((A_1 + A_2) + A_3) + A_4) + ... = A_0 ?$$

Thanks.