nonuniformWitness_spec — Mathlib

English

Any finpartition P yields IsUniform with ε = 1 when appropriate, reflecting a trivial bound.

Русский

Любое разделение FINPAR Wig даёт однородность при ε = 1, отражая тривиальное ограничение.

LaTeX

$$$\text{IsUniform}(P,G,1)\,\text{holds}.$$$

Lean4

theorem nonuniformWitness_spec (h₁ : s ≠ t) (h₂ : ¬G.IsUniform ε s t) :
    ε ≤ |G.edgeDensity (G.nonuniformWitness ε s t) (G.nonuniformWitness ε t s) - G.edgeDensity s t| :=
  by
  unfold nonuniformWitness
  rcases trichotomous_of WellOrderingRel s t with (lt | rfl | gt)
  · rw [if_pos lt, if_neg (asymm lt)]
    exact G.nonuniformWitnesses_spec h₂
  · cases h₁ rfl
  · rw [if_neg (asymm gt), if_pos gt, edgeDensity_comm, edgeDensity_comm _ s]
    apply G.nonuniformWitnesses_spec fun i => h₂ i.symm

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