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Mathlib.Combinatorics.SimpleGraph.Regularity.Energy

Energy of a partition #

This file defines the energy of a partition.

The energy is the auxiliary quantity that drives the induction process in the proof of Szemerédi's Regularity Lemma. As long as we do not have a suitable equipartition, we will find a new one that has an energy greater than the previous one plus some fixed constant.

References #

[Yaël Dillies, Bhavik Mehta, Formalising Szemerédi’s Regularity Lemma in Lean][srl_itp]

def Finpartition.energy {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] :

The energy of a partition, also known as index. Auxiliary quantity for Szemerédi's regularity lemma.

Equations
  • P.energy G = (uvP.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2) / P.parts.card ^ 2
Instances For
    theorem Finpartition.energy_nonneg {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] :
    0 P.energy G
    theorem Finpartition.energy_le_one {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] :
    P.energy G 1
    @[simp]
    theorem Finpartition.coe_energy {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] {𝕜 : Type u_2} [LinearOrderedField 𝕜] :
    (P.energy G) = (uvP.parts.offDiag, (G.edgeDensity uv.1 uv.2) ^ 2) / P.parts.card ^ 2