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Refinement between Lists of lists

Consider the following lists of lists L1 = [[0,1,2],[1,2],[2,2]] and L2 = [[0,1],[2],[1,2],[1,2]]. We say that L2 is a refinement of L1. How to check whether a list of lists is a refinement of another list of lists in Sage.

In the case of set partitions, we have the option refinement. But I need to work with the multisets and its multi partitions. So I am using lists and there is no refinement option for lists of lists.

Kindly help me with how to implement this.

Thank you.