`BookProof.ChapterAttentionOutputVariance.sum_dist_sq_expand` {p : Fin m → ℝ} (hp : ∑ j, p j = 1) (v : Fin m → E) (c : E) : ∑ j, p j * ‖v j - c‖ ^ 2 = (∑ j, p j * ‖v j‖ ^ 2) - 2 *

Proved

BookProof.ChapterAttentionOutputVariance.sum_dist_sq_expand

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutputVariance.

BookProof.ChapterAttentionOutputVariance.sum_dist_sq_expand {p : Fin m → ℝ} (hp : ∑ j, p j = 1) (v : Fin m → E) (c : E) : ∑ j, p j * ‖v j - c‖ ^ 2 = (∑ j, p j * ‖v j‖ ^ 2) - 2 * ⟪observableExpectation p v, c⟫ + ‖c‖ ^ 2

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

Preamble

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


open scoped BigOperators RealInnerProductSpace

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness

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

Formal statement

theorem BookProof.ChapterAttentionOutputVariance.sum_dist_sq_expand {p : Fin m → ℝ} (hp : ∑ j, p j = 1) (v : Fin m → E) (c : E) :
    ∑ j, p j * ‖v j - c‖ ^ 2
      = (∑ j, p j * ‖v j‖ ^ 2) - 2 * ⟪observableExpectation p v, c⟫ + ‖c‖ ^ 2 := by sorry

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