Assume and are disjoint. Suppose and , , , . We would try
but this will not always work because of restrictions on where , , , and are defined.
Given any vector , let be the subvector of elements of which are congruent to mod . Given any permutation of let .
Let be a permutation which sorts into nondescending order. Let . Then is nondescending for every .
Assume is a suitable origin such that and , , , . Let
Let and . Assume is nondescending for every .
For any given , plot the elements of and as a function of position. Call the resulting monotonic polygonal curves and . For example, we might get this picture:
To avoid and having elements in common as we apply operators to we may proceed left to right where lies below and right to left where lies above or on .
Let be a permutation of that in every plot of and for every selects the elements of from left to right where lies below and selects the elements of from right to left where lies above or on . Then we should apply operators to in the order , , . That is,
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