`BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le` (beta : ℝ) {s : Fin m → ℝ} {v : Fin m → E} {C D : ℝ} (hv : ∀ j, ‖v j‖ ≤ C) (hs : ∀ j, |s j - meanScore beta...

Proved

BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionResponse.

BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le (beta : ℝ) {s : Fin m → ℝ} {v : Fin m → E} {C D : ℝ} (hv : ∀ j, ‖v j‖ ≤ C) (hs : ∀ j, |s j - meanScore beta s| ≤ D) (i : Fin m) : ‖scoreValueCovariance beta s v‖ ≤ C * D

Formalization note: Lean 4 identifier BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le.

Preamble

-- Generated from ChapterAttentionResponse.lean — theorem BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionResponse
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxFluctuation
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionResponse


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionResponse.norm_scoreValueCovariance_le (beta : ℝ) {s : Fin m → ℝ} {v : Fin m → E} {C D : ℝ}
    (hv : ∀ j, ‖v j‖ ≤ C) (hs : ∀ j, |s j - meanScore beta s| ≤ D) (i : Fin m) :
    ‖scoreValueCovariance beta s v‖ ≤ C * D := by sorry

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