`BookProof.ChapterLayerNorm.variance_smul` (a : ℝ) (x : Fin d → ℝ) : variance (fun i => a * x i) = a ^ 2 * variance x

Proved

BookProof.ChapterLayerNorm.variance_smul

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLayerNorm.

BookProof.ChapterLayerNorm.variance_smul (a : ℝ) (x : Fin d → ℝ) : variance (fun i => a * x i) = a ^ 2 * variance x

Formalization note: Lean 4 identifier BookProof.ChapterLayerNorm.variance_smul.

Preamble

-- Generated from ChapterLayerNorm.lean — theorem BookProof.ChapterLayerNorm.variance_smul
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterLayerNorm
import Definitions.Def_ChapterTotalVariance
open ChapterTotalVariance
open BookProof.ChapterLayerNorm


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {d : ℕ}

Formal statement

theorem BookProof.ChapterLayerNorm.variance_smul (a : ℝ) (x : Fin d → ℝ) :
    variance (fun i => a * x i) = a ^ 2 * variance x := by sorry

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