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

Proved

BookProof.ChapterAttentionOutputVariance.outputVariance_eq_sub

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutputVariance.

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

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

Preamble

-- Generated from ChapterAttentionOutputVariance.lean — theorem BookProof.ChapterAttentionOutputVariance.outputVariance_eq_sub
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.outputVariance_eq_sub {p : Fin m → ℝ} (hp : ∑ j, p j = 1) (v : Fin m → E) :
    outputVariance p v = (∑ j, p j * ‖v j‖ ^ 2) - ‖observableExpectation p v‖ ^ 2 := by sorry

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