Prove2Me

`BookProof.ChapterLayerNorm.abs_inner_layerNorm_le` (hd : 0 < d) {x y : Fin d → ℝ} (hx : 0 < variance x) (hy : 0 < variance y) : |∑ i, layerNorm x i * layerNorm y i| ≤ (d : ℝ)

Proved
BookProof.ChapterLayerNorm.abs_inner_layerNorm_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLayerNorm.

BookProof.ChapterLayerNorm.abs_inner_layerNorm_le (hd : 0 < d) {x y : Fin d → ℝ} (hx : 0 < variance x) (hy : 0 < variance y) : |∑ i, layerNorm x i * layerNorm y i| ≤ (d : ℝ)

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

Preamble
-- Generated from ChapterLayerNorm.lean — theorem BookProof.ChapterLayerNorm.abs_inner_layerNorm_le
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.abs_inner_layerNorm_le (hd : 0 < d) {x y : Fin d → ℝ} (hx : 0 < variance x)
    (hy : 0 < variance y) :
    |∑ i, layerNorm x i * layerNorm y i| ≤ (d : ℝ) := by sorry

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