`BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval` {k : Fin m → EuclideanSpace ℝ (Fin n)} {v : Fin m → E} {r C eps : ℝ} (hr : 0 < r) (hsep : keysSeparated k r) (hv :

Proved

BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCapacity.

BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval {k : Fin m → EuclideanSpace ℝ (Fin n)} {v : Fin m → E} {r C eps : ℝ} (hr : 0 < r) (hsep : keysSeparated k r) (hv : ∀ l, ‖v l‖ ≤ C) (heps : 0 < eps) : ∃ B : ℝ, 0 ≤ B ∧ ∀ b, B ≤ b → ∀ i : Fin m, ‖headOutput b (distScore (k i) k) v - v i‖ ≤ eps

Formalization note: Lean 4 identifier BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval.

Preamble

-- Generated from ChapterAttentionCapacity.lean — theorem BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionCapacity
import Definitions.Def_ChapterAttentionOutput
open BookProof.ChapterAttentionCapacity


open scoped BigOperators

open Filter Topology

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionOutput

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

Formal statement

theorem BookProof.ChapterAttentionCapacity.exists_beta_forall_retrieval {k : Fin m → EuclideanSpace ℝ (Fin n)}
    {v : Fin m → E} {r C eps : ℝ} (hr : 0 < r) (hsep : keysSeparated k r)
    (hv : ∀ l, ‖v l‖ ≤ C) (heps : 0 < eps) :
    ∃ B : ℝ, 0 ≤ B ∧ ∀ b, B ≤ b → ∀ i : Fin m,
      ‖headOutput b (distScore (k i) k) v - v i‖ ≤ eps := by sorry

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