`BookProof.ChapterAttentionOutput.headOutput_zero` (s : Fin m → ℝ) (v : Fin m → E) : headOutput 0 s v = ((m : ℝ))⁻¹ • ∑ j, v j

Proved

BookProof.ChapterAttentionOutput.headOutput_zero

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutput.

BookProof.ChapterAttentionOutput.headOutput_zero (s : Fin m → ℝ) (v : Fin m → E) : headOutput 0 s v = ((m : ℝ))⁻¹ • ∑ j, v j

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

Preamble

-- Generated from ChapterAttentionOutput.lean — theorem BookProof.ChapterAttentionOutput.headOutput_zero
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionOutput
import Definitions.Def_ChapterSoftmaxSharpness
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.headOutput_zero (s : Fin m → ℝ) (v : Fin m → E) :
    headOutput 0 s v = ((m : ℝ))⁻¹ • ∑ j, v j := by sorry

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