English
Given a partition P of s and a count h, there is a partition equitabilise h which is an equitable partition of s with prescribed counts of part sizes.
Русский
Для данного разбиения P отреза s существует эквитабилитированное разбиение с заданными размерами частей.
LaTeX
$$$\\text{equitabilise} :\\; Finpartition\\ s \\to\\ Finpartition\\ s$ существует с указанными характеристиками по размерам частей (площадки, константы).$$
Lean4
/-- Given a partition `P` of `s`, as well as a proof that `a * m + b * (m + 1) = #s`, build a
new partition `Q` of `s` where each part has size `m` or `m + 1`, every part of `P` is the union of
parts of `Q` plus at most `m` extra elements, there are `b` parts of size `m + 1` and (provided
`m > 0`, because a partition does not have parts of size `0`) there are `a` parts of size `m` and
hence `a + b` parts in total. -/
noncomputable def equitabilise : Finpartition s :=
(P.equitabilise_aux h).choose
