Lax Lean Archive

Bounds for partition energy

Lax18.PartitionEnergyBounds · concepts/Lax18/PartitionEnergyBounds.lean · lax-18

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    Natural Language Statement

    Theorem

    The weighted mean-square density of every finite graph partition lies in the interval from zero to one. This boundedness forces the energy-increment process to terminate after a number of steps depending only on the regularity parameter.

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    1import Lax18.PartitionEnergy
    2
    3/-!
    4---
    5title: Bounds for partition energy
    6type: theorem
    7---
    8The weighted mean-square density of every finite graph partition lies in the
    9interval from zero to one. This boundedness forces the energy-increment
    10process to terminate after a number of steps depending only on the regularity
    11parameter.
    12-/
    13
    14namespace Lax18.PartitionEnergyBounds
    15
    16open Lax18.FiniteGraphPartitions
    17open Lax18.PartitionEnergy
    18
    19universe u
    20
    21/-- Partition energy lies between zero and one. -/
    22axiom partitionEnergy_mem_unitInterval :
    23 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    24 (G : SimpleGraph V) (P : VertexPartition V),
    25 0 ≤ partitionEnergy G P ∧ partitionEnergy G P ≤ 1
    26
    27end Lax18.PartitionEnergyBounds
    28
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