`BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query` {n : ℕ} (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ) (hk : ∀ l, ‖k

Proved

BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionEntropy.

BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query {n : ℕ} (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ) (hk : ∀ l, ‖k l‖ = r) (j : Fin m) (hmax : ∀ l, l ≠ j → (inner ℝ q (k l) : ℝ) < inner ℝ q (k j)) : Tendsto (fun c : ℝ => shannonEntropy (fun l => bornWeight (c • q) k l)) atTop (𝓝 0)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query.

Preamble

-- Generated from ChapterAttentionEntropy.lean — theorem BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query
import Mathlib
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxBorn
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionEntropy


open scoped BigOperators

noncomputable section


open Filter Topology BookProof.ChapterSoftmaxBorn BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionEntropy.tendsto_shannonEntropy_bornWeight_smul_query {n : ℕ}
    (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (r : ℝ)
    (hk : ∀ l, ‖k l‖ = r) (j : Fin m)
    (hmax : ∀ l, l ≠ j → (inner ℝ q (k l) : ℝ) < inner ℝ q (k j)) :
    Tendsto (fun c : ℝ => shannonEntropy (fun l => bornWeight (c • q) k l)) atTop (𝓝 0) := by sorry

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