`BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin` {delta : ℝ} (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m) (hmargin : ∀ l, l ≠

Proved

BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionRetrieval.

BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin {delta : ℝ} (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m) (hmargin : ∀ l, l ≠ j → ‖q - k j‖ ^ 2 + delta ≤ ‖q - k l‖ ^ 2) : 1 - ((m : ℝ) - 1) * Real.exp (-delta) ≤ bornWeight q k j

Formalization note: Lean 4 identifier BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin.

Preamble

-- Generated from ChapterAttentionRetrieval.lean — theorem BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionRetrieval
import Definitions.Def_ChapterSoftmaxBorn
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionRetrieval


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

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

Formal statement

theorem BookProof.ChapterAttentionRetrieval.bornWeight_ge_of_dist_margin {delta : ℝ} (q : EuclideanSpace ℝ (Fin n))
    (k : Fin m → EuclideanSpace ℝ (Fin n)) (j : Fin m)
    (hmargin : ∀ l, l ≠ j → ‖q - k j‖ ^ 2 + delta ≤ ‖q - k l‖ ^ 2) :
    1 - ((m : ℝ) - 1) * Real.exp (-delta) ≤ bornWeight q k j := by sorry

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