`BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq` (beta : ℝ) (s x : Fin m → ℝ) (i : Fin m) : weightedVar beta s x = ∑ j, scoreSoftmax beta s j * (x j - ∑ l, scoreSoftmax bet

Proved

BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxJacobian.

BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq (beta : ℝ) (s x : Fin m → ℝ) (i : Fin m) : weightedVar beta s x = ∑ j, scoreSoftmax beta s j * (x j - ∑ l, scoreSoftmax beta s l * x l) ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq.

Preamble

-- Generated from ChapterSoftmaxJacobian.lean — theorem BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterSoftmaxJacobian
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxJacobian


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxJacobian.weightedVar_eq_sum_sq (beta : ℝ) (s x : Fin m → ℝ) (i : Fin m) :
    weightedVar beta s x
      = ∑ j, scoreSoftmax beta s j * (x j - ∑ l, scoreSoftmax beta s l * x l) ^ 2 := by sorry

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