`BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput` (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (r : ℝ) (hnorm : ∀ l, ‖k l‖ =

Proved

BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutput.

BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (r : ℝ) (hnorm : ∀ l, ‖k l‖ = r) : attentionOutput q k v = headOutput 2 (fun l => (inner ℝ q (k l) : ℝ)) v

Formalization note: Lean 4 identifier BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput.

Preamble

-- Generated from ChapterAttentionOutput.lean — theorem BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionOutput
import Definitions.Def_ChapterObservableExpectation
import Definitions.Def_ChapterSoftmaxBorn
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterObservableExpectation
open BookProof.ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionOutput


open scoped BigOperators

open Filter Topology

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

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

Formal statement

theorem BookProof.ChapterAttentionOutput.attentionOutput_eq_headOutput (q : EuclideanSpace ℝ (Fin n))
    (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (r : ℝ)
    (hnorm : ∀ l, ‖k l‖ = r) :
    attentionOutput q k v = headOutput 2 (fun l => (inner ℝ q (k l) : ℝ)) v := by sorry

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