Increment partition for Szemerédi Regularity Lemma #
In the proof of Szemerédi Regularity Lemma, we need to partition each part of a starting partition to increase the energy. This file defines the partition obtained by gluing the parts partitions together (the increment partition) and shows that the energy globally increases.
This entire file is internal to the proof of Szemerédi Regularity Lemma.
Main declarations #
SzemerediRegularity.increment: The increment partition.
SzemerediRegularity.card_increment: The increment partition is much bigger than the original, but by a controlled amount.
SzemerediRegularity.energy_increment: The increment partition has energy greater than the original by a known (small) fixed amount.
Once ported to mathlib4, this file will be a great golfing ground for Heather's new tactic
[Yaël Dillies, Bhavik Mehta, Formalising Szemerédi’s Regularity Lemma in Lean][srl_itp]
The increment partition in Szemerédi's Regularity Lemma.
If an equipartition is not uniform, then the increment partition is a (much bigger) equipartition
with a slightly higher energy. This is helpful since the energy is bounded by a constant (see
Finpartition.energy_le_one), so this process eventually terminates and yields a
not-too-big uniform equipartition.
The increment partition has a prescribed (very big) size in terms of the original partition.
The increment partition has energy greater than the original one by a known fixed amount.