`BookProof.ChapterLayerNorm.variance_layerNorm` (hd : 0 < d) {x : Fin d → ℝ} (hx : 0 < variance x) : variance (layerNorm x) = 1

Proved

BookProof.ChapterLayerNorm.variance_layerNorm

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLayerNorm.

BookProof.ChapterLayerNorm.variance_layerNorm (hd : 0 < d) {x : Fin d → ℝ} (hx : 0 < variance x) : variance (layerNorm x) = 1

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

Preamble

-- Generated from ChapterLayerNorm.lean — theorem BookProof.ChapterLayerNorm.variance_layerNorm
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_layerNorm (hd : 0 < d) {x : Fin d → ℝ} (hx : 0 < variance x) :
    variance (layerNorm x) = 1 := by sorry

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