`BookProof.ChapterLayerNorm.layerNorm_smul_pos` {a : ℝ} (ha : 0 < a) (x : Fin d → ℝ) (i : Fin d) : layerNorm (fun i => a * x i) i = layerNorm x i

Proved

BookProof.ChapterLayerNorm.layerNorm_smul_pos

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLayerNorm.

BookProof.ChapterLayerNorm.layerNorm_smul_pos {a : ℝ} (ha : 0 < a) (x : Fin d → ℝ) (i : Fin d) : layerNorm (fun i => a * x i) i = layerNorm x i

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

Preamble

-- Generated from ChapterLayerNorm.lean — theorem BookProof.ChapterLayerNorm.layerNorm_smul_pos
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.layerNorm_smul_pos {a : ℝ} (ha : 0 < a) (x : Fin d → ℝ) (i : Fin d) :
    layerNorm (fun i => a * x i) i = layerNorm x i := by sorry

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