Lax Lean Archive

Monotonicity of partition energy under refinement

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

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    Natural Language Statement

    Theorem

    If one graph partition refines another, then its weighted mean-square density is at least that of the coarser partition. This is the finite conditional variance inequality underlying the energy-increment proof.

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    1import Lax18.PartitionEnergy
    2
    3/-!
    4---
    5title: Monotonicity of partition energy under refinement
    6type: theorem
    7---
    8If one graph partition refines another, then its weighted mean-square density
    9is at least that of the coarser partition. This is the finite conditional
    10variance inequality underlying the energy-increment proof.
    11-/
    12
    13namespace Lax18.PartitionEnergyMonotonicity
    14
    15open Lax18.FiniteGraphPartitions
    16open Lax18.PartitionEnergy
    17
    18universe u
    19
    20/-- Refining a partition cannot decrease its energy. -/
    21axiom partitionEnergy_mono_of_refines :
    22 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    23 (G : SimpleGraph V) (P Q : VertexPartition V),
    24 Refines Q P → partitionEnergy G P ≤ partitionEnergy G Q
    25
    26end Lax18.PartitionEnergyMonotonicity
    27
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