`BookProof.ChapterAttentionOutputVariance.outputVariance_eq_zero_iff_of_pos` {p : Fin m → ℝ} (hp0 : ∀ j, 0 < p j) (v : Fin m → E) : outputVariance p v = 0 ↔ ∀ j, v j = observableEx

Proved

BookProof.ChapterAttentionOutputVariance.outputVariance_eq_zero_iff_of_pos

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionOutputVariance.

BookProof.ChapterAttentionOutputVariance.outputVariance_eq_zero_iff_of_pos {p : Fin m → ℝ} (hp0 : ∀ j, 0 < p j) (v : Fin m → E) : outputVariance p v = 0 ↔ ∀ j, v j = observableExpectation p v

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

Preamble

-- Generated from ChapterAttentionOutputVariance.lean — theorem BookProof.ChapterAttentionOutputVariance.outputVariance_eq_zero_iff_of_pos
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_zero_iff_of_pos {p : Fin m → ℝ} (hp0 : ∀ j, 0 < p j)
    (v : Fin m → E) :
    outputVariance p v = 0 ↔ ∀ j, v j = observableExpectation p v := by sorry

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