`BookProof.ChapterAttentionOutputVariance.headOutput_minimizes` (beta : ℝ) (s : Fin m → ℝ) (v : Fin m → E) (i : Fin m) (c : E) : ∑ j, scoreSoftmax beta s j * ‖v j - headOutput beta

Proved

BookProof.ChapterAttentionOutputVariance.headOutput_minimizes

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutputVariance.

BookProof.ChapterAttentionOutputVariance.headOutput_minimizes (beta : ℝ) (s : Fin m → ℝ) (v : Fin m → E) (i : Fin m) (c : E) : ∑ j, scoreSoftmax beta s j * ‖v j - headOutput beta s v‖ ^ 2 ≤ ∑ j, scoreSoftmax beta s j * ‖v j - c‖ ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterAttentionOutputVariance.headOutput_minimizes.

Preamble

-- Generated from ChapterAttentionOutputVariance.lean — theorem BookProof.ChapterAttentionOutputVariance.headOutput_minimizes
import Definitions.Def_ChapterObservableExpectation
import Mathlib
import Definitions.Def_ChapterAttentionOutputVariance
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionOutputVariance


open scoped BigOperators RealInnerProductSpace

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionOutput

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

Formal statement

theorem BookProof.ChapterAttentionOutputVariance.headOutput_minimizes (beta : ℝ) (s : Fin m → ℝ) (v : Fin m → E) (i : Fin m) (c : E) :
    ∑ j, scoreSoftmax beta s j * ‖v j - headOutput beta s v‖ ^ 2
      ≤ ∑ j, scoreSoftmax beta s j * ‖v j - c‖ ^ 2 := by sorry

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