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Lax18.EnergyIncrement

The energy-increment refinement lemma

concepts/Lax18/EnergyIncrement.lean · lax-18

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    Theorem

    This is the standard energy-increment step. If an equitable partition P is not ε-regular, witnesses for all irregular pairs simultaneously split its classes. The resulting refinement has at most k·2^k classes and raises the energy by at least ε⁵/4. The constant is deliberately conservative; what matters for the regularity lemma is a positive increment depending only on ε.

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    1import Lax18.PartitionEnergy
    2
    3/-!
    4---
    5title: The energy-increment refinement lemma
    6type: theorem
    7---
    8This is the standard energy-increment step. If an equitable partition P is
    9not ε-regular, witnesses for all
    10irregular pairs simultaneously split its classes. The resulting refinement
    11has at most k·2^k classes and raises the energy by at least ε⁵/4.
    12The constant is deliberately conservative; what matters
    13for the regularity lemma is a positive increment depending only on
    14ε.
    15-/
    16
    17namespace Lax18.EnergyIncrement
    18
    19open Lax18.FiniteGraphPartitions
    20open Lax18.PartitionEnergy
    21open Lax18.RegularPartitions
    22
    23universe u
    24
    25/-- An irregular equitable partition admits a bounded refinement with a
    26definite energy increment. -/
    27axiom energy_increment_refinement :
    28 ∀ (ε : ℝ),
    29 0 < ε →
    30 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    31 (G : SimpleGraph V) (P : VertexPartition V),
    32 P.Equitable →
    33 ¬ IsRegularPartition G ε P →
    34 ∃ Q : VertexPartition V,
    35 Refines Q P ∧
    36 P.partCount ≤ Q.partCount ∧
    37 Q.partCount ≤ P.partCount * 2 ^ P.partCount ∧
    38 partitionEnergy G P + ε ^ 5 / 4 ≤
    39 partitionEnergy G Q
    40
    41end Lax18.EnergyIncrement
    42
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