`BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff` (beta : ℝ) (s : Fin m → ℝ) : varScore beta s = 0 ↔ ∀ l, s l = meanScore beta s

Proved

BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxFluctuation.

BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff (beta : ℝ) (s : Fin m → ℝ) : varScore beta s = 0 ↔ ∀ l, s l = meanScore beta s

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff.

Preamble

-- Generated from ChapterSoftmaxFluctuation.lean — theorem BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSoftmaxFluctuation
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxFluctuation


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxFluctuation.varScore_eq_zero_iff (beta : ℝ) (s : Fin m → ℝ) :
    varScore beta s = 0 ↔ ∀ l, s l = meanScore beta s := by sorry

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